First slide

Theory of equations

Question

The sum of the roots of the equation $2^{33 x - 2} + 2^{11 x + 2} = 2^{22 x + 1} + 1$ is

Moderate

A

$\frac{1}{11}$
B

$\frac{2}{11}$
C

$\frac{3}{11}$
D

$\frac{4}{11}$

Solution

Let, $2^{11 x} = t$ given equation reduces to
$\frac{t^{3}}{4} + 4 t = 2 t^{2} + 1$
$\begin{aligned} \Rightarrow t^{3} - 8 t^{2} + 16 t - 4 = 0 \Rightarrow t_{1} \cdot t_{2} \cdot t_{3} = 4 \\ \Rightarrow 2^{11 x_{1}} \cdot 2^{11 x_{2}} \cdot 2^{11 x_{3}} = 4 \Rightarrow 2^{11 (x_{1} + x_{2} + x_{3})} = 2^{2} \\ \Rightarrow 11 (x_{1} + x_{2} + x_{3}) = 2 \\ ∴ x_{1} + x_{2} + x_{3} = \frac{2}{11} \end{aligned}$

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