The number of real roots of the equation $5 + | 2^{x} - 1 | = 2^{x} (2^{x} - 2)$ is

4

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Solution

The correct option is B $1$
According to question,
$5 + | 2^{x} - 1 | = 2^{x} (2^{x} - 2)$

Case I
$2^{x} \geq 1$
$\Rightarrow 5 + 2^{x} - 1 = 2^{x} (2^{x} - 2)$
Let $2^{x} = t$
$\Rightarrow 5 + t - 1 = t (t - 2)$
$= t^{2} - 3 t - 4 = 0$
$\Rightarrow t = 4$ or $- 1$ (invalid) $(∵ 2^{x} \neq - 1)$
$\Rightarrow 2^{x} = 4$
$\Rightarrow x = 2$
$∴$ only one solution

Case II
$2^{x} < 1$
$\Rightarrow 5 + 1 - 2^{x} = 2^{x} (2^{x} - 2)$
Let $2^{x} = t$
$\Rightarrow 5 + 1 - t = t (t - 2)$
$\Rightarrow t^{2} - t - 6 = 0$
$\Rightarrow (t - 3) (t - 2) = 0$
$\Rightarrow t = 3, - 2$
$2^{x} = 3$ (invalid as $2^{x} < 1$ ), or $2^{x} = - 2$ (invalid)

Therefore, number of real roots is one.

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The number of real roots of the equation 5+|2x−1|=2x(2x−2) is

The number of real roots of the equation $5 + | 2^{x} - 1 | = 2^{x} (2^{x} - 2)$ is