Solve By Completing The Square.${ X^2 - 24x = 7 } W R I T E Y O U R A N S W E R S A S I N T E G E R S , P R O P E R O R I M P R O P E R F R A C T I O N S I N S I M P L E S T F O R M , O R D E C I M A L S R O U N D E D T O T H E N E A R E S T H U N D R E D T H . Write Your Answers As Integers, Proper Or Improper Fractions In Simplest Form, Or Decimals Rounded To The Nearest Hundredth. W R I T Eyo U R An S W Ers A S In T E G Ers , P Ro P Eror Im P Ro P Er F R A C T I O N S In S Im Pl Es T F Or M , Or D Ec Ima L Sro U N D E D T O T H E N E A Res T H U N D Re D T H . { X = \square \} Or ${ X = \square }$

Mar 11, 2025 by ADMIN 610 views

**Solve by Completing the Square**

What is Completing the Square?

Completing the square is a mathematical technique used to solve quadratic equations of the form ax^2 + bx + c = 0. It involves manipulating the equation to express it in the form (x + d)^2 = e, where d and e are constants. This method is particularly useful for solving quadratic equations that cannot be factored easily.

The Process of Completing the Square

To complete the square, we follow these steps:

Write the equation in the form ax^2 + bx + c = 0. In this case, the equation is x^2 - 24x = 7.
Move the constant term to the right-hand side. This gives us x^2 - 24x - 7 = 0.
Take half of the coefficient of the x-term and square it. The coefficient of the x-term is -24, so half of it is -12. Squaring -12 gives us 144.
Add 144 to both sides of the equation. This gives us x^2 - 24x + 144 = 7 + 144.
Simplify the right-hand side. This gives us x^2 - 24x + 144 = 151.
Write the left-hand side as a perfect square. This gives us (x - 12)^2 = 151.

Solving for x

Now that we have the equation in the form (x - 12)^2 = 151, we can solve for x. To do this, we take the square root of both sides of the equation.

x - 12 = ±√151

x = 12 ± √151

x ≈ 12 ± 12.25

x ≈ 24.25 or x ≈ -0.25

Therefore, the solutions to the equation x^2 - 24x = 7 are x ≈ 24.25 and x ≈ -0.25.

Why is Completing the Square Useful?

Completing the square is a useful technique for solving quadratic equations because it allows us to express the equation in a form that is easy to work with. It is particularly useful for solving equations that cannot be factored easily, and it can be used to solve equations with complex roots.

Real-World Applications of Completing the Square

Completing the square has many real-world applications, including:

Physics: Completing the square is used to solve equations of motion, such as the equation of a projectile under the influence of gravity.
Engineering: Completing the square is used to solve equations of electrical circuits, such as the equation of a resistor-capacitor circuit.
Computer Science: Completing the square is used to solve equations of algorithms, such as the equation of a sorting algorithm.

Conclusion

In conclusion, completing the square is a powerful technique for solving quadratic equations. It involves manipulating the equation to express it in the form (x + d)^2 = e, where d and e are constants. This method is particularly useful for solving equations that cannot be factored easily, and it has many real-world applications in physics, engineering, and computer science.

References

"Algebra" by Michael Artin: This book provides a comprehensive introduction to algebra, including the technique of completing the square.
"Calculus" by Michael Spivak: This book provides a comprehensive introduction to calculus, including the technique of completing the square.
"Mathematics for Computer Science" by Eric Lehman: This book provides a comprehensive introduction to mathematics for computer science, including the technique of completing the square.

Q: What is completing the square?

A: Completing the square is a mathematical technique used to solve quadratic equations of the form ax^2 + bx + c = 0. It involves manipulating the equation to express it in the form (x + d)^2 = e, where d and e are constants.

Q: When should I use completing the square?

A: You should use completing the square when you have a quadratic equation that cannot be factored easily. This method is particularly useful for solving equations with complex roots.

Q: How do I complete the square?

A: To complete the square, follow these steps:

Write the equation in the form ax^2 + bx + c = 0.
Move the constant term to the right-hand side.
Take half of the coefficient of the x-term and square it.
Add the result to both sides of the equation.
Simplify the right-hand side.
Write the left-hand side as a perfect square.

Q: What if I have a quadratic equation with a negative coefficient?

A: If you have a quadratic equation with a negative coefficient, you can still complete the square. Simply follow the same steps as before, but be careful when taking the square root of both sides of the equation.

Q: Can I use completing the square to solve quadratic equations with complex roots?

A: Yes, you can use completing the square to solve quadratic equations with complex roots. However, you will need to use the complex conjugate of the root to find the other root.

Q: How do I find the complex conjugate of a root?

A: To find the complex conjugate of a root, simply change the sign of the imaginary part of the root. For example, if the root is 3 + 4i, the complex conjugate is 3 - 4i.

Q: Can I use completing the square to solve quadratic equations with rational roots?

A: Yes, you can use completing the square to solve quadratic equations with rational roots. However, you will need to use the rational root theorem to find the possible rational roots of the equation.

Q: What is the rational root theorem?

A: The rational root theorem states that if a rational number p/q is a root of the polynomial ax^2 + bx + c = 0, then p must be a factor of c and q must be a factor of a.

Q: Can I use completing the square to solve quadratic equations with irrational roots?

A: Yes, you can use completing the square to solve quadratic equations with irrational roots. However, you will need to use the quadratic formula to find the roots of the equation.

Q: What is the quadratic formula?

A: The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a.

Q: Can I use completing the square to solve quadratic equations with repeated roots?

A: Yes, you can use completing the square to solve quadratic equations with repeated roots. However, you will need to use the fact that the repeated root is equal to the negative of the coefficient of the x-term divided by the coefficient of the x^2-term.

Q: What are some common mistakes to avoid when completing the square?

A: Some common mistakes to avoid when completing the square include:

Not moving the constant term to the right-hand side of the equation.
Not taking half of the coefficient of the x-term and squaring it.
Not adding the result to both sides of the equation.
Not simplifying the right-hand side of the equation.
Not writing the left-hand side as a perfect square.

Q: How do I know if I have completed the square correctly?

A: To check if you have completed the square correctly, simply plug the solution back into the original equation and check if it is true. If it is true, then you have completed the square correctly.

Q: Can I use completing the square to solve systems of quadratic equations?

A: Yes, you can use completing the square to solve systems of quadratic equations. However, you will need to use the method of substitution or elimination to solve the system.

Q: What are some real-world applications of completing the square?

A: Some real-world applications of completing the square include:

Solving equations of motion in physics.
Solving equations of electrical circuits in engineering.
Solving equations of algorithms in computer science.

Q: Can I use completing the square to solve quadratic equations with complex coefficients?

A: Yes, you can use completing the square to solve quadratic equations with complex coefficients. However, you will need to use the complex conjugate of the root to find the other root.

Q: How do I find the complex conjugate of a root with complex coefficients?

A: To find the complex conjugate of a root with complex coefficients, simply change the sign of the imaginary part of the root. For example, if the root is 3 + 4i, the complex conjugate is 3 - 4i.