Question : If $\left(w+\frac{1}{w}\right)=6$, then what will be the value of $\left(w-\frac{1}{w}\right)?$
Option 1: $4 \sqrt{2}$
Option 2: $\sqrt{2}$
Option 3: $3 \sqrt{2}$
Option 4: $2 \sqrt{2}$
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Correct Answer: $4 \sqrt{2}$
Solution :
Given: $\left(w+\frac{1}{w}\right)=6$
Squaring both sides, we have,
⇒ $\left(w+\frac{1}{w}\right)^{2}=6^{2}$
⇒ $w^{2}+\frac{1}{w^{2}}+2=36$
⇒ $w^{2}+\frac{1}{w^{2}}=34$
⇒ $w^{2}+\frac{1}{w^{2}}-2=34-2$
⇒ $(w-\frac{1}{w})^{2}=32$
⇒ $(w-\frac{1}{w})=\sqrt{32}$
$\therefore(w-\frac{1}{w})=4\sqrt{2}$
Hence, the correct answer is $4\sqrt{2}$.
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