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Introduction In this article, we will delve into the world of algebra and solve a quadratic equation involving a square root. The given equation is 8 x + 65 = x + 10 \sqrt{8x + 65} = x + 10 8 x + 65 β = x + 10 x x x x x x
Understanding the Equation The given equation involves a square root, which can be challenging to work with. However, we can start by isolating the square root term on one side of the equation. This will allow us to eliminate the square root and solve for x x x
Step 1: Isolate the Square Root Term To isolate the square root term, we can subtract x + 10 x + 10 x + 10
8 x + 65 β ( x + 10 ) = 0 \sqrt{8x + 65} - (x + 10) = 0
8 x + 65 β β ( x + 10 ) = 0
Step 2: Simplify the Equation We can simplify the equation by combining like terms. This gives us:
8 x + 65 β x β 10 = 0 \sqrt{8x + 65} - x - 10 = 0
8 x + 65 β β x β 10 = 0
Step 3: Square Both Sides To eliminate the square root, we can square both sides of the equation. This gives us:
( 8 x + 65 β x β 10 ) 2 = 0 (\sqrt{8x + 65} - x - 10)^2 = 0
( 8 x + 65 β β x β 10 ) 2 = 0
Expanding the squared term, we get:
8 x + 65 β 2 x 8 x + 65 + x 2 + 100 β 20 x = 0 8x + 65 - 2x\sqrt{8x + 65} + x^2 + 100 - 20x = 0
8 x + 65 β 2 x 8 x + 65 β + x 2 + 100 β 20 x = 0
Step 4: Simplify the Equation We can simplify the equation by combining like terms. This gives us:
x 2 β 18 x + 165 β 2 x 8 x + 65 = 0 x^2 - 18x + 165 - 2x\sqrt{8x + 65} = 0
x 2 β 18 x + 165 β 2 x 8 x + 65 β = 0
Step 5: Move the Square Root Term To isolate the square root term, we can move it to the right-hand side of the equation. This gives us:
x 2 β 18 x + 165 = 2 x 8 x + 65 x^2 - 18x + 165 = 2x\sqrt{8x + 65}
x 2 β 18 x + 165 = 2 x 8 x + 65 β
Step 6: Square Both Sides Again To eliminate the square root, we can square both sides of the equation again. This gives us:
( x 2 β 18 x + 165 ) 2 = 4 x 2 ( 8 x + 65 ) (x^2 - 18x + 165)^2 = 4x^2(8x + 65)
( x 2 β 18 x + 165 ) 2 = 4 x 2 ( 8 x + 65 )
Expanding both sides, we get:
x 4 β 36 x 3 + 729 x 2 β 6120 x + 27225 = 32 x 3 + 260 x 2 x^4 - 36x^3 + 729x^2 - 6120x + 27225 = 32x^3 + 260x^2
x 4 β 36 x 3 + 729 x 2 β 6120 x + 27225 = 32 x 3 + 260 x 2
Step 7: Simplify the Equation We can simplify the equation by combining like terms. This gives us:
x 4 β 68 x 3 + 469 x 2 β 6120 x + 27225 = 0 x^4 - 68x^3 + 469x^2 - 6120x + 27225 = 0
x 4 β 68 x 3 + 469 x 2 β 6120 x + 27225 = 0
Solving the Equation The resulting equation is a quartic equation, which can be challenging to solve. However, we can use various algebraic techniques to simplify the equation and solve for x x x
Step 1: Factor the Equation We can factor the equation by grouping terms. This gives us:
( x 2 β 68 x + 469 ) ( x 2 β 6120 x + 27225 ) = 0 (x^2 - 68x + 469)(x^2 - 6120x + 27225) = 0
( x 2 β 68 x + 469 ) ( x 2 β 6120 x + 27225 ) = 0
Step 2: Solve for x We can solve for x x x
x 2 β 68 x + 469 = 0 x^2 - 68x + 469 = 0
x 2 β 68 x + 469 = 0
x 2 β 6120 x + 27225 = 0 x^2 - 6120x + 27225 = 0
x 2 β 6120 x + 27225 = 0
Solving the Quadratic Equations We can solve the quadratic equations using various algebraic techniques.
Step 1: Use the Quadratic Formula The quadratic formula is:
x = β b Β± b 2 β 4 a c 2 a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
x = 2 a β b Β± b 2 β 4 a c β β
We can use the quadratic formula to solve the quadratic equations.
Step 2: Simplify the Solutions We can simplify the solutions by combining like terms. This gives us:
x = 68 Β± 4624 β 7548 2 x = \frac{68 \pm \sqrt{4624 - 7548}}{2}
x = 2 68 Β± 4624 β 7548 β β
x = 68 Β± β 2924 2 x = \frac{68 \pm \sqrt{-2924}}{2}
x = 2 68 Β± β 2924 β β
Conclusion The resulting equation is a quartic equation, which can be challenging to solve. However, we can use various algebraic techniques to simplify the equation and solve for x x x
x = 68 Β± β 2924 2 x = \frac{68 \pm \sqrt{-2924}}{2}
x = 2 68 Β± β 2924 β β
Note that the solutions involve complex numbers, which may not be the desired outcome. However, the solutions provide a starting point for further analysis and simplification.
Final Answer The final answer is:
x = \frac{68 \pm \sqrt{-2924}}{2}$<br/>
# **Frequently Asked Questions (FAQs) About Solving the Equation for x**
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Q: What is the given equation? A: The given equation is 8 x + 65 = x + 10 \sqrt{8x + 65} = x + 10 8 x + 65 β = x + 10
Q: What is the goal of solving the equation? A: The goal of solving the equation is to isolate the variable x x x
Q: What algebraic techniques were used to solve the equation? A: Various algebraic techniques were used to solve the equation, including:
Isolating the square root term
Squaring both sides of the equation
Simplifying the equation
Factoring the equation
Using the quadratic formula
Q: What is the resulting equation after squaring both sides? A: The resulting equation after squaring both sides is:
( x 2 β 18 x + 165 ) 2 = 4 x 2 ( 8 x + 65 ) < / s p a n > < / p > < h 2 > < s t r o n g > Q : W h a t i s t h e r e s u l t i n g e q u a t i o n a f t e r s i m p l i f y i n g ? < / s t r o n g > < / h 2 > < h r > < p > A : T h e r e s u l t i n g e q u a t i o n a f t e r s i m p l i f y i n g i s : < / p > < p c l a s s = β² k a t e x β b l o c k β² > < s p a n c l a s s = " k a t e x β d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m s u p > < m i > x < / m i > < m n > 4 < / m n > < / m s u p > < m o > β < / m o > < m n > 68 < / m n > < m s u p > < m i > x < / m i > < m n > 3 < / m n > < / m s u p > < m o > + < / m o > < m n > 469 < / m n > < m s u p > < m i > x < / m i > < m n > 2 < / m n > < / m s u p > < m o > β < / m o > < m n > 6120 < / m n > < m i > x < / m i > < m o > + < / m o > < m n > 27225 < / m n > < m o > = < / m o > < m n > 0 < / m n > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > x 4 β 68 x 3 + 469 x 2 β 6120 x + 27225 = 0 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.9474 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t β t " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " > < s p a n s t y l e = " t o p : β 3.113 e m ; m a r g i n β r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 4 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > β < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.9474 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 68 < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t β t " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " > < s p a n s t y l e = " t o p : β 3.113 e m ; m a r g i n β r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 3 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.9474 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 469 < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t β t " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " > < s p a n s t y l e = " t o p : β 3.113 e m ; m a r g i n β r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > β < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 6120 < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 27225 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 0 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < h 2 > < s t r o n g > Q : W h a t t y p e o f e q u a t i o n i s t h e r e s u l t i n g e q u a t i o n ? < / s t r o n g > < / h 2 > < h r > < p > A : T h e r e s u l t i n g e q u a t i o n i s a q u a r t i c e q u a t i o n . < / p > < h 2 > < s t r o n g > Q : H o w w a s t h e q u a r t i c e q u a t i o n s i m p l i f i e d ? < / s t r o n g > < / h 2 > < h r > < p > A : T h e q u a r t i c e q u a t i o n w a s s i m p l i f i e d b y f a c t o r i n g i t i n t o t w o q u a d r a t i c e q u a t i o n s : < / p > < p c l a s s = β² k a t e x β b l o c k β² > < s p a n c l a s s = " k a t e x β d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m s u p > < m i > x < / m i > < m n > 2 < / m n > < / m s u p > < m o > β < / m o > < m n > 68 < / m n > < m i > x < / m i > < m o > + < / m o > < m n > 469 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m s u p > < m i > x < / m i > < m n > 2 < / m n > < / m s u p > < m o > β < / m o > < m n > 6120 < / m n > < m i > x < / m i > < m o > + < / m o > < m n > 27225 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > = < / m o > < m n > 0 < / m n > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > ( x 2 β 68 x + 469 ) ( x 2 β 6120 x + 27225 ) = 0 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.1141 e m ; v e r t i c a l β a l i g n : β 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t β t " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " > < s p a n s t y l e = " t o p : β 3.113 e m ; m a r g i n β r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > β < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 68 < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.1141 e m ; v e r t i c a l β a l i g n : β 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 469 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t β t " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " > < s p a n s t y l e = " t o p : β 3.113 e m ; m a r g i n β r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > β < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 6120 < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l β a l i g n : β 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 27225 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 0 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < h 2 > < s t r o n g > Q : W h a t a r e t h e s o l u t i o n s t o t h e e q u a t i o n ? < / s t r o n g > < / h 2 > < h r > < p > A : T h e s o l u t i o n s t o t h e e q u a t i o n a r e : < / p > < p c l a s s = β² k a t e x β b l o c k β² > < s p a n c l a s s = " k a t e x β d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m i > x < / m i > < m o > = < / m o > < m f r a c > < m r o w > < m n > 68 < / m n > < m o > Β± < / m o > < m s q r t > < m r o w > < m o > β < / m o > < m n > 2924 < / m n > < / m r o w > < / m s q r t > < / m r o w > < m n > 2 < / m n > < / m f r a c > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > x = 68 Β± β 2924 2 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.2286 e m ; v e r t i c a l β a l i g n : β 0.686 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.5426 e m ; " > < s p a n s t y l e = " t o p : β 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c β l i n e " s t y l e = " b o r d e r β b o t t o m β w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 68 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Β± < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d s q r t " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8656 e m ; " > < s p a n c l a s s = " s v g β a l i g n " s t y l e = " t o p : β 3 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " s t y l e = " p a d d i n g β l e f t : 0.833 e m ; " > < s p a n c l a s s = " m o r d " > β < / s p a n > < s p a n c l a s s = " m o r d " > 2924 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 2.8256 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " h i d e β t a i l " s t y l e = " m i n β w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " > < s v g x m l n s = " h t t p : / / w w w . w 3. o r g / 2000 / s v g " w i d t h = " 400 e m " h e i g h t = " 1.08 e m " v i e w B o x = " 004000001080 " p r e s e r v e A s p e c t R a t i o = " x M i n Y M i n s l i c e " > < p a t h d = " M 95 , 702 c β 2.7 , 0 , β 7.17 , β 2.7 , β 13.5 , β 8 c β 5.8 , β 5.3 , β 9.5 , β 10 , β 9.5 , β 14 c 0 , β 2 , 0.3 , β 3.3 , 1 , β 4 c 1.3 , β 2.7 , 23.83 , β 20.7 , 67.5 , β 54 c 44.2 , β 33.3 , 65.8 , β 50.3 , 66.5 , β 51 c 1.3 , β 1.3 , 3 , β 2 , 5 , β 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , β 71 , 104 , β 213 c 68.7 , β 142 , 137.5 , β 285 , 206.5 , β 429 c 69 , β 144 , 104.5 , β 217.7 , 106.5 , β 221 l 0 β 0 c 5.3 , β 9.3 , 12 , β 14 , 20 , β 14 H 400000 v 40 H 845.2724 s β 225.272 , 467 , β 225.272 , 467 s β 235 , 486 , β 235 , 486 c β 2.7 , 4.7 , β 9 , 7 , β 19 , 7 c β 6 , 0 , β 10 , β 1 , β 12 , β 3 s β 194 , β 422 , β 194 , β 422 s β 65 , 47 , β 65 , 47 z M 83480 h 400000 v 40 h β 400000 z " / > < / s v g > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1744 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < h 2 > < s t r o n g > Q : A r e t h e s o l u t i o n s r e a l o r c o m p l e x ? < / s t r o n g > < / h 2 > < h r > < p > A : T h e s o l u t i o n s a r e c o m p l e x n u m b e r s . < / p > < h 2 > < s t r o n g > Q : W h y a r e t h e s o l u t i o n s c o m p l e x ? < / s t r o n g > < / h 2 > < h r > < p > A : T h e s o l u t i o n s a r e c o m p l e x b e c a u s e t h e e q u a t i o n i n v o l v e s a s q u a r e r o o t , w h i c h c a n l e a d t o c o m p l e x n u m b e r s w h e n s q u a r e d . < / p > < h 2 > < s t r o n g > Q : W h a t i s t h e f i n a l a n s w e r ? < / s t r o n g > < / h 2 > < h r > < p > A : T h e f i n a l a n s w e r i s : < / p > < p c l a s s = β² k a t e x β b l o c k β² > < s p a n c l a s s = " k a t e x β d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m i > x < / m i > < m o > = < / m o > < m f r a c > < m r o w > < m n > 68 < / m n > < m o > Β± < / m o > < m s q r t > < m r o w > < m o > β < / m o > < m n > 2924 < / m n > < / m r o w > < / m s q r t > < / m r o w > < m n > 2 < / m n > < / m f r a c > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > x = 68 Β± β 2924 2 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.2286 e m ; v e r t i c a l β a l i g n : β 0.686 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.5426 e m ; " > < s p a n s t y l e = " t o p : β 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c β l i n e " s t y l e = " b o r d e r β b o t t o m β w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 68 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Β± < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d s q r t " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8656 e m ; " > < s p a n c l a s s = " s v g β a l i g n " s t y l e = " t o p : β 3 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " s t y l e = " p a d d i n g β l e f t : 0.833 e m ; " > < s p a n c l a s s = " m o r d " > β < / s p a n > < s p a n c l a s s = " m o r d " > 2924 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 2.8256 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " h i d e β t a i l " s t y l e = " m i n β w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " > < s v g x m l n s = " h t t p : / / w w w . w 3. o r g / 2000 / s v g " w i d t h = " 400 e m " h e i g h t = " 1.08 e m " v i e w B o x = " 004000001080 " p r e s e r v e A s p e c t R a t i o = " x M i n Y M i n s l i c e " > < p a t h d = " M 95 , 702 c β 2.7 , 0 , β 7.17 , β 2.7 , β 13.5 , β 8 c β 5.8 , β 5.3 , β 9.5 , β 10 , β 9.5 , β 14 c 0 , β 2 , 0.3 , β 3.3 , 1 , β 4 c 1.3 , β 2.7 , 23.83 , β 20.7 , 67.5 , β 54 c 44.2 , β 33.3 , 65.8 , β 50.3 , 66.5 , β 51 c 1.3 , β 1.3 , 3 , β 2 , 5 , β 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , β 71 , 104 , β 213 c 68.7 , β 142 , 137.5 , β 285 , 206.5 , β 429 c 69 , β 144 , 104.5 , β 217.7 , 106.5 , β 221 l 0 β 0 c 5.3 , β 9.3 , 12 , β 14 , 20 , β 14 H 400000 v 40 H 845.2724 s β 225.272 , 467 , β 225.272 , 467 s β 235 , 486 , β 235 , 486 c β 2.7 , 4.7 , β 9 , 7 , β 19 , 7 c β 6 , 0 , β 10 , β 1 , β 12 , β 3 s β 194 , β 422 , β 194 , β 422 s β 65 , 47 , β 65 , 47 z M 83480 h 400000 v 40 h β 400000 z " / > < / s v g > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1744 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < h 2 > < s t r o n g > Q : W h a t i s t h e s i g n i f i c a n c e o f t h e f i n a l a n s w e r ? < / s t r o n g > < / h 2 > < h r > < p > A : T h e f i n a l a n s w e r p r o v i d e s a s t a r t i n g p o i n t f o r f u r t h e r a n a l y s i s a n d s i m p l i f i c a t i o n o f t h e e q u a t i o n . < / p > < h 2 > < s t r o n g > Q : C a n t h e e q u a t i o n b e s o l v e d u s i n g o t h e r m e t h o d s ? < / s t r o n g > < / h 2 > < h r > < p > A : Y e s , t h e e q u a t i o n c a n b e s o l v e d u s i n g o t h e r m e t h o d s , s u c h a s n u m e r i c a l m e t h o d s o r a p p r o x i m a t i o n t e c h n i q u e s . < / p > < h 2 > < s t r o n g > Q : W h a t a r e t h e l i m i t a t i o n s o f t h e c u r r e n t s o l u t i o n ? < / s t r o n g > < / h 2 > < h r > < p > A : T h e c u r r e n t s o l u t i o n i n v o l v e s c o m p l e x n u m b e r s , w h i c h m a y n o t b e t h e d e s i r e d o u t c o m e . A d d i t i o n a l l y , t h e s o l u t i o n m a y n o t b e e x a c t , b u t r a t h e r a n a p p r o x i m a t i o n . < / p > < h 2 > < s t r o n g > Q : W h a t a r e t h e n e x t s t e p s i n s o l v i n g t h e e q u a t i o n ? < / s t r o n g > < / h 2 > < h r > < p > A : T h e n e x t s t e p s i n s o l v i n g t h e e q u a t i o n i n v o l v e f u r t h e r a n a l y s i s a n d s i m p l i f i c a t i o n o f t h e e q u a t i o n , a s w e l l a s e x p l o r i n g a l t e r n a t i v e m e t h o d s f o r s o l v i n g t h e e q u a t i o n . < / p > (x^2 - 18x + 165)^2 = 4x^2(8x + 65)
</span></p>
<h2><strong>Q: What is the resulting equation after simplifying?</strong></h2>
<hr>
<p>A: The resulting equation after simplifying is:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>β</mo><mn>68</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>469</mn><msup><mi>x</mi><mn>2</mn></msup><mo>β</mo><mn>6120</mn><mi>x</mi><mo>+</mo><mn>27225</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x^4 - 68x^3 + 469x^2 - 6120x + 27225 = 0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">β</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mord">68</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mord">469</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">β</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6120</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">27225</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>
<h2><strong>Q: What type of equation is the resulting equation?</strong></h2>
<hr>
<p>A: The resulting equation is a quartic equation.</p>
<h2><strong>Q: How was the quartic equation simplified?</strong></h2>
<hr>
<p>A: The quartic equation was simplified by factoring it into two quadratic equations:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>β</mo><mn>68</mn><mi>x</mi><mo>+</mo><mn>469</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>β</mo><mn>6120</mn><mi>x</mi><mo>+</mo><mn>27225</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(x^2 - 68x + 469)(x^2 - 6120x + 27225) = 0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">β</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">68</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord">469</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">β</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6120</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">27225</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>
<h2><strong>Q: What are the solutions to the equation?</strong></h2>
<hr>
<p>A: The solutions to the equation are:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>68</mn><mo>Β±</mo><msqrt><mrow><mo>β</mo><mn>2924</mn></mrow></msqrt></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">x = \frac{68 \pm \sqrt{-2924}}{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2286em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5426em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">68</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Β±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8656em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">β</span><span class="mord">2924</span></span></span><span style="top:-2.8256em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.1744em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<h2><strong>Q: Are the solutions real or complex?</strong></h2>
<hr>
<p>A: The solutions are complex numbers.</p>
<h2><strong>Q: Why are the solutions complex?</strong></h2>
<hr>
<p>A: The solutions are complex because the equation involves a square root, which can lead to complex numbers when squared.</p>
<h2><strong>Q: What is the final answer?</strong></h2>
<hr>
<p>A: The final answer is:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>68</mn><mo>Β±</mo><msqrt><mrow><mo>β</mo><mn>2924</mn></mrow></msqrt></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">x = \frac{68 \pm \sqrt{-2924}}{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2286em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5426em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">68</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Β±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8656em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">β</span><span class="mord">2924</span></span></span><span style="top:-2.8256em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.1744em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<h2><strong>Q: What is the significance of the final answer?</strong></h2>
<hr>
<p>A: The final answer provides a starting point for further analysis and simplification of the equation.</p>
<h2><strong>Q: Can the equation be solved using other methods?</strong></h2>
<hr>
<p>A: Yes, the equation can be solved using other methods, such as numerical methods or approximation techniques.</p>
<h2><strong>Q: What are the limitations of the current solution?</strong></h2>
<hr>
<p>A: The current solution involves complex numbers, which may not be the desired outcome. Additionally, the solution may not be exact, but rather an approximation.</p>
<h2><strong>Q: What are the next steps in solving the equation?</strong></h2>
<hr>
<p>A: The next steps in solving the equation involve further analysis and simplification of the equation, as well as exploring alternative methods for solving the equation.</p>
( x 2 β 18 x + 165 ) 2 = 4 x 2 ( 8 x + 65 ) < / s p an >< / p >< h 2 >< s t ro n g > Q : Wha t i s t h eres u lt in g e q u a t i o na f t ers im pl i f y in g ? < / s t ro n g >< / h 2 >< h r >< p > A : T h eres u lt in g e q u a t i o na f t ers im pl i f y in g i s :< / p >< p c l a ss = β² ka t e x β b l oc k β² >< s p an c l a ss = " ka t e x β d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< m s u p >< mi > x < / mi >< mn > 4 < / mn >< / m s u p >< m o > β < / m o >< mn > 68 < / mn >< m s u p >< mi > x < / mi >< mn > 3 < / mn >< / m s u p >< m o > + < / m o >< mn > 469 < / mn >< m s u p >< mi > x < / mi >< mn > 2 < / mn >< / m s u p >< m o > β < / m o >< mn > 6120 < / mn >< mi > x < / mi >< m o > + < / m o >< mn > 27225 < / mn >< m o >=< / m o >< mn > 0 < / mn >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > x 4 β 68 x 3 + 469 x 2 β 6120 x + 27225 = 0 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.9474 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t β t " >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " >< s p an s t y l e = " t o p : β 3.113 e m ; ma r g in β r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 4 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > β < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.9474 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 68 < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t β t " >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " >< s p an s t y l e = " t o p : β 3.113 e m ; ma r g in β r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 3 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.9474 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 469 < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t β t " >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " >< s p an s t y l e = " t o p : β 3.113 e m ; ma r g in β r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > β < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 6120 < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 27225 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 0 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< h 2 >< s t ro n g > Q : Wha tt y p eo f e q u a t i o ni s t h eres u lt in g e q u a t i o n ? < / s t ro n g >< / h 2 >< h r >< p > A : T h eres u lt in g e q u a t i o ni s a q u a r t i ce q u a t i o n . < / p >< h 2 >< s t ro n g > Q : Ho ww a s t h e q u a r t i ce q u a t i o n s im pl i f i e d ? < / s t ro n g >< / h 2 >< h r >< p > A : T h e q u a r t i ce q u a t i o n w a ss im pl i f i e d b y f a c t or in g i t in t o tw o q u a d r a t i ce q u a t i o n s :< / p >< p c l a ss = β² ka t e x β b l oc k β² >< s p an c l a ss = " ka t e x β d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< m os t re t c h y = " f a l se " > ( < / m o >< m s u p >< mi > x < / mi >< mn > 2 < / mn >< / m s u p >< m o > β < / m o >< mn > 68 < / mn >< mi > x < / mi >< m o > + < / m o >< mn > 469 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< m s u p >< mi > x < / mi >< mn > 2 < / mn >< / m s u p >< m o > β < / m o >< mn > 6120 < / mn >< mi > x < / mi >< m o > + < / m o >< mn > 27225 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m o >=< / m o >< mn > 0 < / mn >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > ( x 2 β 68 x + 469 ) ( x 2 β 6120 x + 27225 ) = 0 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.1141 e m ; v er t i c a l β a l i g n : β 0.25 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t β t " >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " >< s p an s t y l e = " t o p : β 3.113 e m ; ma r g in β r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > β < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 68 < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.1141 e m ; v er t i c a l β a l i g n : β 0.25 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 469 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t β t " >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " >< s p an s t y l e = " t o p : β 3.113 e m ; ma r g in β r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > β < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 6120 < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l β a l i g n : β 0.25 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 27225 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 0 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< h 2 >< s t ro n g > Q : Wha t a re t h eso l u t i o n s t o t h ee q u a t i o n ? < / s t ro n g >< / h 2 >< h r >< p > A : T h eso l u t i o n s t o t h ee q u a t i o na re :< / p >< p c l a ss = β² ka t e x β b l oc k β² >< s p an c l a ss = " ka t e x β d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< mi > x < / mi >< m o >=< / m o >< m f r a c >< m ro w >< mn > 68 < / mn >< m o > Β± < / m o >< m s q r t >< m ro w >< m o > β < / m o >< mn > 2924 < / mn >< / m ro w >< / m s q r t >< / m ro w >< mn > 2 < / mn >< / m f r a c >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > x = 2 68 Β± β 2924 β β < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.2286 e m ; v er t i c a l β a l i g n : β 0.686 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.5426 e m ; " >< s p an s t y l e = " t o p : β 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c β l in e " s t y l e = " b or d er β b o tt o m β w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 68 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Β± < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " m or d s q r t " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8656 e m ; " >< s p an c l a ss = " s vg β a l i g n " s t y l e = " t o p : β 3 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " s t y l e = " p a dd in g β l e f t : 0.833 e m ; " >< s p an c l a ss = " m or d " > β < / s p an >< s p an c l a ss = " m or d " > 2924 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 2.8256 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " hi d e β t ai l " s t y l e = " min β w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " >< s vgx m l n s = " h ttp : // www . w 3. or g /2000/ s vg " w i d t h = "400 e m " h e i g h t = "1.08 e m " v i e wB o x = "004000001080" p reser v e A s p ec tR a t i o = " x M inY M in s l i ce " >< p a t h d = " M 95 , 702 c β 2.7 , 0 , β 7.17 , β 2.7 , β 13.5 , β 8 c β 5.8 , β 5.3 , β 9.5 , β 10 , β 9.5 , β 14 c 0 , β 2 , 0.3 , β 3.3 , 1 , β 4 c 1.3 , β 2.7 , 23.83 , β 20.7 , 67.5 , β 54 c 44.2 , β 33.3 , 65.8 , β 50.3 , 66.5 , β 51 c 1.3 , β 1.3 , 3 , β 2 , 5 , β 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , β 71 , 104 , β 213 c 68.7 , β 142 , 137.5 , β 285 , 206.5 , β 429 c 69 , β 144 , 104.5 , β 217.7 , 106.5 , β 221 l 0 β 0 c 5.3 , β 9.3 , 12 , β 14 , 20 , β 14 H 400000 v 40 H 845.2724 s β 225.272 , 467 , β 225.272 , 467 s β 235 , 486 , β 235 , 486 c β 2.7 , 4.7 , β 9 , 7 , β 19 , 7 c β 6 , 0 , β 10 , β 1 , β 12 , β 3 s β 194 , β 422 , β 194 , β 422 s β 65 , 47 , β 65 , 47 z M 83480 h 400000 v 40 h β 400000 z "/ >< / s vg >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1744 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< h 2 >< s t ro n g > Q : A re t h eso l u t i o n sre a l orco m pl e x ? < / s t ro n g >< / h 2 >< h r >< p > A : T h eso l u t i o n s a reco m pl e x n u mb ers . < / p >< h 2 >< s t ro n g > Q : Wh y a re t h eso l u t i o n sco m pl e x ? < / s t ro n g >< / h 2 >< h r >< p > A : T h eso l u t i o n s a reco m pl e x b ec a u se t h ee q u a t i o nin v o l v es a s q u a reroo t , w hi c h c an l e a d t oco m pl e x n u mb ers w h e n s q u a re d . < / p >< h 2 >< s t ro n g > Q : Wha t i s t h e f ina l an s w er ? < / s t ro n g >< / h 2 >< h r >< p > A : T h e f ina l an s w er i s :< / p >< p c l a ss = β² ka t e x β b l oc k β² >< s p an c l a ss = " ka t e x β d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< mi > x < / mi >< m o >=< / m o >< m f r a c >< m ro w >< mn > 68 < / mn >< m o > Β± < / m o >< m s q r t >< m ro w >< m o > β < / m o >< mn > 2924 < / mn >< / m ro w >< / m s q r t >< / m ro w >< mn > 2 < / mn >< / m f r a c >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > x = 2 68 Β± β 2924 β β < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.2286 e m ; v er t i c a l β a l i g n : β 0.686 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.5426 e m ; " >< s p an s t y l e = " t o p : β 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c β l in e " s t y l e = " b or d er β b o tt o m β w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 68 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Β± < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " m or d s q r t " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8656 e m ; " >< s p an c l a ss = " s vg β a l i g n " s t y l e = " t o p : β 3 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " s t y l e = " p a dd in g β l e f t : 0.833 e m ; " >< s p an c l a ss = " m or d " > β < / s p an >< s p an c l a ss = " m or d " > 2924 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 2.8256 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " hi d e β t ai l " s t y l e = " min β w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " >< s vgx m l n s = " h ttp : // www . w 3. or g /2000/ s vg " w i d t h = "400 e m " h e i g h t = "1.08 e m " v i e wB o x = "004000001080" p reser v e A s p ec tR a t i o = " x M inY M in s l i ce " >< p a t h d = " M 95 , 702 c β 2.7 , 0 , β 7.17 , β 2.7 , β 13.5 , β 8 c β 5.8 , β 5.3 , β 9.5 , β 10 , β 9.5 , β 14 c 0 , β 2 , 0.3 , β 3.3 , 1 , β 4 c 1.3 , β 2.7 , 23.83 , β 20.7 , 67.5 , β 54 c 44.2 , β 33.3 , 65.8 , β 50.3 , 66.5 , β 51 c 1.3 , β 1.3 , 3 , β 2 , 5 , β 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , β 71 , 104 , β 213 c 68.7 , β 142 , 137.5 , β 285 , 206.5 , β 429 c 69 , β 144 , 104.5 , β 217.7 , 106.5 , β 221 l 0 β 0 c 5.3 , β 9.3 , 12 , β 14 , 20 , β 14 H 400000 v 40 H 845.2724 s β 225.272 , 467 , β 225.272 , 467 s β 235 , 486 , β 235 , 486 c β 2.7 , 4.7 , β 9 , 7 , β 19 , 7 c β 6 , 0 , β 10 , β 1 , β 12 , β 3 s β 194 , β 422 , β 194 , β 422 s β 65 , 47 , β 65 , 47 z M 83480 h 400000 v 40 h β 400000 z "/ >< / s vg >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1744 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< h 2 >< s t ro n g > Q : Wha t i s t h es i g ni f i c an ceo f t h e f ina l an s w er ? < / s t ro n g >< / h 2 >< h r >< p > A : T h e f ina l an s w er p ro v i d es a s t a r t in g p o in t f or f u r t h er ana l ys i s an d s im pl i f i c a t i o n o f t h ee q u a t i o n . < / p >< h 2 >< s t ro n g > Q : C an t h ee q u a t i o nb eso l v e d u s in g o t h er m e t h o d s ? < / s t ro n g >< / h 2 >< h r >< p > A : Y es , t h ee q u a t i o n c anb eso l v e d u s in g o t h er m e t h o d s , s u c ha s n u m er i c a l m e t h o d sor a pp ro x ima t i o n t ec hni q u es . < / p >< h 2 >< s t ro n g > Q : Wha t a re t h e l imi t a t i o n so f t h ec u rre n t so l u t i o n ? < / s t ro n g >< / h 2 >< h r >< p > A : T h ec u rre n t so l u t i o nin v o l v esco m pl e x n u mb ers , w hi c hma y n o t b e t h e d es i re d o u t co m e . A dd i t i o na ll y , t h eso l u t i o nma y n o t b ee x a c t , b u t r a t h er ana pp ro x ima t i o n . < / p >< h 2 >< s t ro n g > Q : Wha t a re t h e n e x t s t e p s in so l v in g t h ee q u a t i o n ? < / s t ro n g >< / h 2 >< h r >< p > A : T h e n e x t s t e p s in so l v in g t h ee q u a t i o nin v o l v e f u r t h er ana l ys i s an d s im pl i f i c a t i o n o f t h ee q u a t i o n , a s w e ll a se x pl or in g a lt er na t i v e m e t h o d s f orso l v in g t h ee q u a t i o n . < / p >