Question

The roots of the equation $3^{x+2}+3^{-x}$ = 10 are
(a) 2, 0
(b) −2, 0
(c) 3, −1
(d) −3, 1

Answer 1

(b) −2, 0

$$\begin{gathered} \text{Given:} \\ {3}^{x+2}+3^{-x}=10 \\ {\text{Let 3}}^x\text{be equal to}y. \\ \text{Then, the equation can be rewritten as:} \\ {\text{3}}^2\text{×}{3}^x+\frac{1}{3^x}=10 \\ \Rightarrow 9y+\frac{1}{y}=10 \\ \text{⇒}\frac{9y^2+1}{y}=10 \\ \text{⇒ 9}{y}^2-10y+1=0 \\ \text{⇒ 9}{y}^2-9y-y+1=0 \\ \Rightarrow 9y(y-1)-(y-1)=0 \\ \text{⇒}(y-1)(9y-1)=0 \\ \text{⇒}y=1\text{or}y=\frac{1}{9}=\frac{1}{3^2} \\ {\text{⇒ 3}}^x=3^0{\text{or 3}}^x=3^{-2} \\ \text{⇒}x=0\text{or}x=-2 \end{gathered}$$