If \(a+\displaystyle\frac{1}{a+7}=5\), then find the value of \((a+7)^2+\displaystyle\frac{1}{(a+7)^2}\).
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 142
Adding 7 to both sides makes the left side of the form \(t+\frac{1}{t}\) with \(t=a+7\); then \(t^2+\frac{1}{t^2}=\left(t+\frac{1}{t}\right)^2-2\).
\((a+7)+\displaystyle\frac{1}{a+7}=5+7=12\)
\((a+7)^2+\displaystyle\frac{1}{(a+7)^2}=12^2-2=144-2=142\)