Question
2. If \( x+\frac{1}{x}=9 \), find \( x^{2}+\frac{1}{x^{2}} \)
Ask by Cross Gonzalez. in Pakistan
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x^{2} + \frac{1}{x^{2}} = 79 \)
Solution
To find \( x^{2} + \frac{1}{x^{2}} \) given that \( x + \frac{1}{x} = 9 \), follow these steps:
1. **Square Both Sides of the Given Equation:**
\[
\left( x + \frac{1}{x} \right)^2 = 9^2
\]
\[
x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 81
\]
2. **Simplify the Equation:**
Notice that \( 2 \cdot x \cdot \frac{1}{x} = 2 \). So,
\[
x^2 + 2 + \frac{1}{x^2} = 81
\]
3. **Isolate \( x^{2} + \frac{1}{x^{2}} \):**
\[
x^2 + \frac{1}{x^2} = 81 - 2
\]
\[
x^2 + \frac{1}{x^2} = 79
\]
**Answer:** \( x^{2} + \frac{1}{x^{2}} = 79 \)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To find \( x^{2}+\frac{1}{x^{2}} \), we can use the identity: \[ x^{2} + \frac{1}{x^{2}} = \left( x + \frac{1}{x} \right)^{2} - 2 \] Given that \( x + \frac{1}{x} = 9 \), we first calculate: \[ \left( x + \frac{1}{x} \right)^{2} = 9^{2} = 81 \] Now, substituting this back into the identity: \[ x^{2} + \frac{1}{x^{2}} = 81 - 2 = 79 \] Thus, the value of \( x^{2} + \frac{1}{x^{2}} \) is \( \boxed{79} \).
