If x + 1/x = √8, find x^2 + 1/x^2.
Understand the Problem
The question is asking for the value of x^2 + 1/x^2 given the equation x + 1/x = √8. To solve this, we can use the identity that relates these two expressions. By squaring both sides of the initial equation, we can derive the desired value.
Answer
The value of $x^2 + \frac{1}{x^2}$ is $6$.
Answer for screen readers
The value of $x^2 + \frac{1}{x^2}$ is $6$.
Steps to Solve
- Square the given equation
We start with the equation
$$ x + \frac{1}{x} = \sqrt{8} $$
Now, we square both sides:
$$ \left(x + \frac{1}{x}\right)^2 = (\sqrt{8})^2 $$
- Expand the left side
Expanding the left side gives us:
$$ x^2 + 2\cdot x\cdot \frac{1}{x} + \frac{1}{x^2} = 8 $$
The term $2\cdot x\cdot \frac{1}{x}$ simplifies to 2:
$$ x^2 + 2 + \frac{1}{x^2} = 8 $$
- Rearrange the equation
Now, we can isolate the terms involving $x^2$ and $\frac{1}{x^2}$:
$$ x^2 + \frac{1}{x^2} = 8 - 2 $$
- Simplify to find the desired value
Now simplify the right side:
$$ x^2 + \frac{1}{x^2} = 6 $$
The value of $x^2 + \frac{1}{x^2}$ is $6$.
More Information
This problem involves manipulating algebraic expressions, and a useful identity is that squaring the sum of a variable and its reciprocal can help relate to the square of their individual terms.
Tips
- Forgetting to square both parts of the equation properly when expanding.
- Miscalculating the final simplified value after rearranging terms.