Question
f \( x-\frac{1}{x}=4 \), prove that \( x^{2}+\frac{1}{x^{2}}=18 \)
Ask by Stephens George. in Bangladesh
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x^{2} + \frac{1}{x^{2}} = 18 \)
Solution
To prove that \( x^{2} + \frac{1}{x^{2}} = 18 \) given that \( x - \frac{1}{x} = 4 \), follow these steps:
### Step 1: Start with the Given Equation
You are given:
\[
x - \frac{1}{x} = 4
\]
### Step 2: Square Both Sides of the Equation
Squaring both sides will help eliminate the fraction and introduce \( x^2 \) and \( \frac{1}{x^2} \), which are present in the expression we need to prove.
\[
\left( x - \frac{1}{x} \right)^2 = 4^2
\]
Compute both sides:
\[
x^2 - 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 16
\]
\[
x^2 - 2 + \frac{1}{x^2} = 16
\]
### Step 3: Simplify the Equation
Add 2 to both sides to isolate \( x^2 + \frac{1}{x^2} \):
\[
x^2 + \frac{1}{x^2} = 16 + 2
\]
\[
x^2 + \frac{1}{x^2} = 18
\]
### Conclusion
Therefore, given that \( x - \frac{1}{x} = 4 \), it follows that:
\[
x^{2} + \frac{1}{x^{2}} = 18
\]
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The Deep Dive
To show that \( x^{2}+\frac{1}{x^{2}}=18 \) given \( f(x) = x - \frac{1}{x} = 4 \), we can start by squaring the initial equation. 1. Square both sides: \[ \left(x - \frac{1}{x}\right)^2 = 4^2 \] This simplifies to: \[ x^2 - 2x\cdot\frac{1}{x} + \frac{1}{x^2} = 16 \] Which further simplifies to: \[ x^2 - 2 + \frac{1}{x^2} = 16 \] 2. Adding 2 to both sides gives: \[ x^2 + \frac{1}{x^2} = 16 + 2 = 18 \] Thus, we have proven that \( x^{2} + \frac{1}{x^{2}} = 18 \).