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  4. The roots of the equation 4x-3 ⋅ 2x+3+128=0 are

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Q. The roots of the equation $4^{x}-3 \cdot 2^{x+3}+128=0$ are

Complex Numbers and Quadratic Equations

Solution:

We have, $4^{x}-3 2^{x+3}+128=0$
$\Rightarrow 2^{2 x}-3 \cdot 2^{x} \cdot 2^{3}+128=0$
$\Rightarrow 2^{2 x}-24 \cdot 2^{x}+128=0$
$\Rightarrow y^{2}-24 y+128=0$ where $2^{x}=y$
$\Rightarrow (y-16)(y-8)=0 $
$\Rightarrow y=16,8$
$\Rightarrow 2^{x}=16$ or $2^{x}=8$
$ \Rightarrow x=4$ or 3