$2^{2 x + 3} = 65 (2^{x} - 1) + 57$

\pm 3

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\pm 1

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\pm 2

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None of these

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Solution

The correct option is C $\pm 3$
Given equation is: $2^{2 x + 3} = 65 (2^{x} - 1) + 57$
$\Rightarrow {(2^{x})}^{2} \times 2^{3} = 65 (2^{x} - 1) + 57$
Let, $2^{x} = y$
$\Rightarrow {8 y}^{2} = 65 y - 65 + 57$
$\Rightarrow {8 y}^{2} - 65 y + 8 = 0$
$\Rightarrow {8 y}^{2} - 64 y - y + 8 = 0$
$\Rightarrow 8 y (y - 8) - 1 (y - 8) = 0$
$\Rightarrow (y - 8) (8 y - 1) = 0$
$\Rightarrow y = 8$ or $y = \frac{1}{8}$
Resubstitute the value of $y = 2^{x}$
$∴ 2^{x} = y = 8 = {(2)}^{3} \Rightarrow x = 3$
$∴ 2^{x} = y = \frac{1}{8} = \frac{1}{2^{3}} = {(- 2)}^{3} \Rightarrow x = - 3$

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