Solve each of the following quadratic equations:
$4^{(x + 1)} + 4^{(1 - x)} = 10$

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Solution

$4^{(x + 1)} + 4^{(1 - x)} = 10$

${4.4}^{x} + {4.4}^{- x} = 10$

${4.4}^{x} + \frac{4}{4^{x}} = 10$

Let $4^{x} = k$ , then our expression becomes:

$4 k + \frac{4}{k} = 10$

$\frac{4 k^{2} + 4}{k} = 10$

$4 k^{2} + 4 = 10 k$

$2 k^{2} + 2 = 5 k$

$2 k^{2} - 5 k + 2 = 0$

$2 k (k - 2) - 1 (k - 2) = 0$

$(2 k - 1) (k - 2) = 0$

$k = \frac{1}{2}, 2$

$4^{x} = \frac{1}{2} o r 4^{x} = 2 (2^{2})^{x} = 2^{- 1} o r (2^{2})^{x} = 2^{1} 2^{2 x} = 2^{- 1} o r 2^{2 x} = 2^{1} 2 x = - 1 o r 2 x = 1 x = \frac{- 1}{2} o r x = \frac{1}{2}$

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