Question : If $5x+\frac{1}{x}=10$, then $x^2+\frac{1}{25x^2}$ is equal to:
Option 1: $2\frac{1}{5}$
Option 2: $3\frac{1}{5}$
Option 3: $3\frac{3}{5}$
Option 4: $2\frac{3}{5}$
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Correct Answer: $3\frac{3}{5}$
Solution :
Given: $5x+\frac{1}{x}=10$
We know that the algebraic identity, $(a+b)^2=a^2+b^2+2ab$.
$5x+\frac{1}{x}=10$
On squaring both sides of the above equation, we get,
$(5x+\frac{1}{x})^2=10^2$
⇒ $25x^2+\frac{1}{x^2}+10=100$
⇒ $25x^2+\frac{1}{x^2}=90$
Divide by 25 on both sides of the above equation, we get,
$x^2+\frac{1}{25x^2}=\frac{90}{25}$
⇒ $x^2+\frac{1}{25x^2}=\frac{18}{5}=3\frac{3}{5}$
Hence, the correct answer is $3\frac{3}{5}$.
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