If \(a-\displaystyle\frac{1}{a-1}=7\), then find the value of \((a-1)^2+\displaystyle\frac{1}{(a-1)^2}\).
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 38
Subtracting 1 from both sides makes the left side of the form \(t-\frac{1}{t}\) with \(t=a-1\); then \(t^2+\frac{1}{t^2}=\left(t-\frac{1}{t}\right)^2+2\).
\((a-1)-\displaystyle\frac{1}{a-1}=7-1=6\)
\((a-1)^2+\displaystyle\frac{1}{(a-1)^2}=6^2+2=36+2=38\)