Question

Solve for $x$:

$8\times 2^{2x}+4\times 2^{x+1}=1+2^x$

• A. $3$
• B. $-3$
• C. $2$
• D. $1$

Answer 1

The correct option is B $-3$

$8\times 2^{2x}+4\times 2^{x+1}=1+2^x$

$\Rightarrow$ $8\times (2^x{)}^2+4\times 2^x\times 2^1=1+2^x$

$\Rightarrow$ Let $2^x=p$

$\Rightarrow$ $8\times p^2+4\times p\times 2^1=1+p$

$\Rightarrow$ $8p^2+8p-p-1=0$

$\Rightarrow$ $8p^2+7p-1=0$

$\Rightarrow$ $8p^2+8p-p-1=0$

$\Rightarrow$ $8p(p+1)-(p+1)=0$

$\Rightarrow$ $(8p-1)(p+1)=0$

$\Rightarrow$ $8p-1=0$

$\Rightarrow$ $p=\frac{1}{8}$

$\Rightarrow$ but $p=2^x$ so, $2^x=\frac{1}{8}$

$\Rightarrow$ $2^x=2^{-3}$

$\therefore$ $x=-3$