Question
Solve the following equations for $x$:
$2^{\text{2x}}-2^{\text{x}+3}+2^4=0$
$2^{\text{2x}}-2^{\text{x}+3}+2^4=0$
✓
Answer
We have,$\Rightarrow2^{\text{2x}}-2^{\text{x}+3}+2^4=0$
$\Rightarrow2^2\text{x}+2^4=2^\text{x}.2^3$
$\Rightarrow\text{Let}\ 2^\text{x}=\text{y}$
$\Rightarrow\text{y}^2+2^4=\text{y}\times2^3$
$\Rightarrow\text{y}^2=8\text{y}+16=0$
$\Rightarrow\text{y}^2-4\text{y}-4\text{y}+16=0$
$\Rightarrow\text{y}(\text{y}-4)-4(\text{y}-4)=0$
$\Rightarrow\text{y}=4$
$\Rightarrow\text{x}^2=2^2$
$\Rightarrow\text{x}=2$
$\Rightarrow2^2\text{x}+2^4=2^\text{x}.2^3$
$\Rightarrow\text{Let}\ 2^\text{x}=\text{y}$
$\Rightarrow\text{y}^2+2^4=\text{y}\times2^3$
$\Rightarrow\text{y}^2=8\text{y}+16=0$
$\Rightarrow\text{y}^2-4\text{y}-4\text{y}+16=0$
$\Rightarrow\text{y}(\text{y}-4)-4(\text{y}-4)=0$
$\Rightarrow\text{y}=4$
$\Rightarrow\text{x}^2=2^2$
$\Rightarrow\text{x}=2$
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