Question
If \( x-\frac{1}{x}=12 \), find \( x^{2}+\frac{1}{x^{2}} \)
Ask by Paul Ayala. in Pakistan
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x^{2} + \frac{1}{x^{2}} = 146 \)
Solution
To find \( x^{2} + \frac{1}{x^{2}} \) given that \( x - \frac{1}{x} = 12 \), we can follow these steps:
1. **Square Both Sides:**
\[
\left(x - \frac{1}{x}\right)^2 = 12^2
\]
\[
x^2 - 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 144
\]
\[
x^2 - 2 + \frac{1}{x^2} = 144
\]
2. **Solve for \( x^2 + \frac{1}{x^2} \):**
\[
x^2 + \frac{1}{x^2} = 144 + 2
\]
\[
x^2 + \frac{1}{x^2} = 146
\]
**Answer:** \( \boxed{146} \)
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Simplify this solution Bonus Knowledge
To solve for \( x^{2}+\frac{1}{x^{2}} \) given that \( x - \frac{1}{x} = 12 \), we can use the identity that relates these quantities. First, we square both sides of the equation \( x - \frac{1}{x} = 12 \): \[ \left(x - \frac{1}{x}\right)^2 = 12^2 \] Expanding the left side, we get: \[ x^2 - 2x\cdot\frac{1}{x} + \frac{1}{x^2} = 144 \] This simplifies to: \[ x^2 - 2 + \frac{1}{x^2} = 144 \] Rearranging this gives us: \[ x^2 + \frac{1}{x^2} = 144 + 2 = 146 \] Thus, the value of \( x^{2}+\frac{1}{x^{2}} \) is \[ \boxed{146} \]
