Question : If $x+\frac{16}{x}=8$, then the value of $x^2+\frac{32}{x^2}$ is:
Option 1: 20
Option 2: 24
Option 3: 16
Option 4: 18
Correct Answer: 18
Solution :
Given: The value of $x+\frac{16}{x}=8$.
We know the algebraic identity, $(a–b)^2=a^2+b^2–2ab$.
$x+\frac{16}{x}=8$
⇒ $x^2+16=8x$
⇒ $x^2–8x+16=0$
⇒ $(x–4)^2=0$
⇒ $x=4,4$
The value of $x^2+\frac{32}{x^2}$ is given as,
⇒ $4^2+\frac{32}{4^2}=16+\frac{32}{16}$
= $16+2=18$
Hence, the correct answer is 18.
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