Question

If $x+\frac{1}{x}=2$ then $x^2+\frac{1}{x^2}=?$

Answer 1

Step 1: Solve the given expression

Given expression, $x+\frac{1}{x}=2$

On squaring both sides, we get;

$$\begin{gathered} {\Rightarrow (x+\frac{1}{x})}^2=2^2 \\ {\Rightarrow (x+\frac{1}{x})}^2=4 \end{gathered}$$

Step 2: Use algebraic identities

Using identity, ${(\mathrm{a}+\mathrm{b})}^2={\mathrm{a}}^2+{\mathrm{b}}^2+2\mathrm{ab}$

$$\begin{gathered} \Rightarrow x^2+{\left(\frac{1}{x}\right)}^2+2\left(x\right)\left(\frac{1}{x}\right)=4 \\ \Rightarrow x^2+\frac{1}{x^2}+2=4 \\ \Rightarrow x^2+\frac{1}{x^2}=4-2 \\ \Rightarrow x^2+\frac{1}{x^2}=2 \end{gathered}$$

Hence, $x^2+\frac{1}{x^2}=2$.