Solve the following quadratic equation: 4^ (x+1) + 4^ (1-x) = 10 - Vidyadip

Question

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

✓

Answer

$4^{(x+1)} + 4^{(1-x)} = 10$
$4^x.4^1 + 4^1.4^{-x} = 10$
$\Rightarrow\text{4y}+\frac{4}{\text{y}}=10$ where $4^x = y$
$\Rightarrow 4y^2 - 10y + 4 = 0$
$\Rightarrow 4y^2 - 8y - 2y + 4 = 0$
$\Rightarrow 4y(y - 2) - 2(y - 2) = 0$
$\Rightarrow (y - 2)(4y - 2) = 0$
$\Rightarrow y - 2 = 0$ or $4y - 2 = 0$
$\Rightarrow y = 2$ or $\text{y}=\frac{2}{4}=\frac{1}2{}$
$\Rightarrow y = 2$ or $\text{y}=\frac{1}{2}$
In case I:
$\Rightarrow 4^x = 2$
$\Rightarrow (2)^{2x} = (2)^1$
$\Rightarrow 2x = 1$
$\Rightarrow\text{x}=\frac{1}{2}$
In case II:
$\Rightarrow4^\text{x}=\frac{1}{2}$
$\Rightarrow(2)^\text{2x}=\Big(\frac{1}{2}\Big)^1$
$\Rightarrow(2)^{\text{2x}}=(2)^{-1}$
$\Rightarrow\text{x}=-\frac{1}{2}$
Hence, $\frac{1}{2},-\frac{1}2{}$ are the roots of given equation.

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