Solve for x:2^ 2x + 2^ x +2 - 4 2^3 = 0 - Vidyadip

Question

Solve for $x:2^{2x} + 2^{x +2} - 4 \times 2^3 = 0$

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Answer

$2^{2x} + 2^{x +2} - 4 x 2^3 = 0$
$\Rightarrow 2^{2x} + 2^{x+2} - 2^2 x 2^3 = 0$
$\Rightarrow 2^{2x} + 2^x . 2^2 - 2^{2+3} = 0 ......($Using $a^m x a^n= a^{m+n})$
$\Rightarrow 2^{2x}+ 2^x . 2^2 - 2^5 = 0$
$\Rightarrow 2^{2x} + 2^x . 4 - 32 = 0$
Put $2^x=t$
So, $2^{2 x }= t ^2$
$2^{2x} + 2^{x+2} - 32 = 0$
becomes $t^2 + 4t - 32 = 0$
$\Rightarrow (t + 8)(t - 4) = 0$
$\Rightarrow t + 8 = 0$ or $t - 4 = 0$
$\Rightarrow t = -8 = 0$ or $t = 4$
$\Rightarrow 2^x = -8$ or $2^x = 4$
$\Rightarrow 2x = -2^3$ or $2^x = 2^2$
Using the second equation $2^x=2^2$, we get $x=2$.

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