Question

$4^{(\mathrm{x}+1)}+4^{(1-x)}=10$

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ {4}^{(x+1)}+4^{(1-x)}=10 \\ \Rightarrow 4^x.4+4^1.\frac{1}{4^x}=10 \\ {\text{Let 4}}^x\text{be}y. \\ \begin{array}{l}\therefore 4y+\frac{4}{y}=10\\ \Rightarrow 4y^2-10y+4=0\end{array} \\ \Rightarrow 4y^2-8y-2y+4=0 \\ \Rightarrow 4y(y-2)-2(y-2)=0 \\ \begin{array}{l}\Rightarrow y=2\text{or}y=\frac{2}{4}=\frac{1}{2}\\ \Rightarrow 4^x=2\mathrm{or}\frac{1}{2}\end{array} \\ \Rightarrow 4^x=2^{2x}=2^1\mathrm{or}{2}^{2x}=2^{-1} \\ \Rightarrow x=\frac{1}{2}\mathrm{or}x=\frac{-1}{2} \\ \text{Hence,}\frac{1}{2}\text{and}\frac{-1}{2}\text{are roots of the given equation}. \end{gathered}$$