If X=3 +√ 8 Then Find X²+1/x²
Solving the Equation: If x=3 +√8 Then Find x²+1/x²
In this article, we will delve into the world of algebra and explore a complex equation involving square roots and fractions. The equation in question is x=3 +√8, and we are tasked with finding the value of x²+1/x². To solve this problem, we will need to employ various mathematical techniques, including algebraic manipulation and the use of the quadratic formula.
Understanding the Equation
The given equation is x=3 +√8. This equation involves a square root, which can be simplified by first finding the value of √8. To do this, we can use the fact that √8 can be expressed as √(4*2), which is equal to 2√2.
import math
x = 3 + math.sqrt(8)
sqrt_8 = math.sqrt(4 * 2)
sqrt_8 = 2 * math.sqrt(2)
Simplifying the Equation
Now that we have simplified the square root, we can rewrite the equation as x=3 +2√2. This equation can be further simplified by combining the like terms.
# Simplify the equation
x = 3 + 2 * math.sqrt(2)
Finding x²
To find the value of x², we can simply square both sides of the equation. This will give us x²=(3 +2√2)².
# Find x²
x_squared = (3 + 2 * math.sqrt(2)) ** 2
Expanding the Equation
To expand the equation, we can use the binomial theorem, which states that (a+b)²=a²+2ab+b². In this case, a=3 and b=2√2.
# Expand the equation
x_squared = 3 ** 2 + 2 * 3 * 2 * math.sqrt(2) + (2 * math.sqrt(2)) ** 2
Simplifying the Equation
Now that we have expanded the equation, we can simplify it by combining the like terms.
# Simplify the equation
x_squared = 9 + 12 * math.sqrt(2) + 8
Finding 1/x²
To find the value of 1/x², we can take the reciprocal of x².
# Find 1/x²
one_over_x_squared = 1 / x_squared
Finding x²+1/x²
To find the value of x²+1/x², we can simply add the two expressions together.
# Find x²+1/x²
x_squared_plus_one_over_x_squared = x_squared + one_over_x_squared
Simplifying the Equation
Now that we have found the value of x²+1/x², we can simplify the equation by combining the like terms.
# Simplify the equation
x_squared_plus_one_over_x_squared = 17 + 12 * math.sqrt(2)
In this article, we have solved the equation x=3 +√8 and found the value of x²+1/x². We have used various mathematical techniques, including algebraic manipulation and the use of the quadratic formula. The final answer is x²+1/x²=17 + 12 * √2.
The final answer is:
Frequently Asked Questions: If x=3 +√8 Then Find x²+1/x²
In our previous article, we solved the equation x=3 +√8 and found the value of x²+1/x². However, we understand that some readers may still have questions about the solution. In this article, we will address some of the most frequently asked questions about the equation and its solution.
Q: What is the value of √8?
A: The value of √8 can be simplified by expressing it as √(4*2), which is equal to 2√2.
Q: How do I simplify the equation x=3 +√8?
A: To simplify the equation, you can combine the like terms. The equation can be rewritten as x=3 +2√2.
Q: How do I find the value of x²?
A: To find the value of x², you can simply square both sides of the equation. This will give you x²=(3 +2√2)².
Q: How do I expand the equation x²=(3 +2√2)²?
A: To expand the equation, you can use the binomial theorem, which states that (a+b)²=a²+2ab+b². In this case, a=3 and b=2√2.
Q: How do I simplify the equation x²=9 +12√2 +8?
A: To simplify the equation, you can combine the like terms. The equation can be rewritten as x²=17 +12√2.
Q: How do I find the value of 1/x²?
A: To find the value of 1/x², you can take the reciprocal of x².
Q: How do I find the value of x²+1/x²?
A: To find the value of x²+1/x², you can simply add the two expressions together.
Q: What is the final answer to the equation x=3 +√8?
A: The final answer to the equation x=3 +√8 is x²+1/x²=17 +12√2.
Q: Can I use a calculator to solve the equation?
A: Yes, you can use a calculator to solve the equation. However, it's always a good idea to understand the steps involved in solving the equation.
Q: Can I use a computer program to solve the equation?
A: Yes, you can use a computer program to solve the equation. However, it's always a good idea to understand the steps involved in solving the equation.
In this article, we have addressed some of the most frequently asked questions about the equation x=3 +√8 and its solution. We hope that this article has been helpful in clarifying any confusion about the solution.
The final answer is: