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f \( x-\frac{1}{x}=4 \), prove that \( x^{2}+\frac{1}{x^{2}}=18 \)

Ask by Stephens George. in Bangladesh
Jan 24,2025

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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 \( x^{2} + \frac{1}{x^{2}} = 18 \).

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