Question

Solve the following equations for x:

(i) $2^{2x}-2^{x+3}+2^4=0$

(ii) $3^{2x+4}+1=2.3^{x+2}$

Answer 1

(i)
$$\begin{gathered} 2^{2x}-2^{x+3}+2^4=0 \\ \Rightarrow {\left(2^x\right)}^2-\left(2^x\times 2^3\right)+{\left(2^2\right)}^2=0 \\ \Rightarrow {\left(2^x\right)}^2-2\times 2^x\times 2^2+{\left(2^2\right)}^2=0 \\ \Rightarrow {\left(2^x-2^2\right)}^2=0 \\ \Rightarrow 2^x-2^2=0 \\ \Rightarrow 2^x=2^2 \\ \Rightarrow x=2 \end{gathered}$$

(ii)
$$\begin{gathered} 3^{2x+4}+1=2.3^{x+2} \\ \Rightarrow {\left(3^{x+2}\right)}^2-2.3^{x+2}+1=0 \\ \Rightarrow {\left(3^{x+2}-1\right)}^2=0 \\ \Rightarrow 3^{x+2}-1=0 \\ \Rightarrow 3^{x+2}=1=3^0 \\ \Rightarrow x+2=0 \\ \Rightarrow x=-2 \end{gathered}$$