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Solving a Quadratic Equation by Completion of Squares Method

Solve for x.2...

Question

Solve for x.
$2^{2 x + 3} = 65 (2^{X} - 2) + 122$

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Solution

$2^{2 x + 3} = {65.2}^{x} - 130 + 122$
$2^{2 x} \times 2^{3} = {65.2}^{x} - 8 L e t 2^{x} = t \Rightarrow t^{2} \times 8 = 64. t - 8$
$\Rightarrow 8 t^{2} - 65 t + 8 = 0 \Rightarrow 8 t^{2} - 64 t - t + 8 = 0 \Rightarrow 8 + (t - 8) - 1 (t - 8) = 0 \Rightarrow (8 t - 1) (t - 8) = 0 \Rightarrow t = \frac{1}{8}, 8 \Rightarrow 2^{x} = \frac{1}{8}, 2^{x} = 8 2^{x} = \frac{1}{2^{3}}, 2^{x} = 2^{3} 2^{x} = 2^{- 3} 2^{x} = 2^{3} \Rightarrow x = 3 2^{x} = 2^{- 3}, x = 3$

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