Question : If $x^2-3.2 x+1=0$ and $x>1$, the value of $x^2-\frac{1}{x^2}$ is:
Option 1: $16.8 \sqrt{0.39}$
Option 2: $12.8 \sqrt{0.39}$
Option 3: $16.8 \sqrt{0.32}$
Option 4: $12.8 \sqrt{0.32}$
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Correct Answer: $12.8 \sqrt{0.39}$
Solution :
Given: $x^2-3.2 x+1=0$ and $x>1$
$x^2-3.2 x+1=0$
Dividing both sides by $x$, we get,
$⇒x-3.2 +\frac{1}{x}=0$
$⇒x+\frac{1}{x}=3.2 $---(1)
Squaring both sides, we get,
$⇒x^2+\frac{1}{x^2}+2=10.24 $
$⇒x^2+\frac{1}{x^2}-2=6.24 $
$⇒(x-\frac{1}{x})^2=6.24 $
$⇒(x-\frac{1}{x})=\sqrt{6.24}$---(2)
Multiplying 1 and 2,
$(x+\frac{1}{x})(x-\frac{1}{x})=3.2×\sqrt{6.24}=3.2×\sqrt{16×0.39}=12.8 \sqrt{0.39}$
Hence, the correct answer is $12.8 \sqrt{0.39}$.
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