Question

Tick (✓) the correct answer:
If $\left(x+\frac{1}{x}\right)=5$, then $\left(x^2+\frac{1}{x^2}\right)=?$
(a) 25
(b) 27
(c) 23
(d) $25\frac{1}{25}$

Answer 1

(c) 23

$$\begin{gathered} \left(x+\frac{1}{x}\right)=5 \\ \Rightarrow Squaringboththesides: \\ \Rightarrow {\left(x+\frac{1}{x}\right)}^2={\left(5\right)}^2 \\ \Rightarrow \left(x^2+\frac{1}{x^2}+2\left(x\right)\left(\frac{1}{x}\right)\right)=25 \\ \Rightarrow \left(x^2+\frac{1}{x^2}\right)+2=25 \\ \Rightarrow \left(x^2+\frac{1}{x^2}\right)=25-2 \\ \Rightarrow \left(x^2+\frac{1}{x^2}\right)=23 \end{gathered}$$