First slide

Theory of equations

Question

Solution set of the equation ${(x - 1)}^{4} + {(x - 5)}^{4} = 82$ is

Difficult

A

${2}$
B

${4}$
C

${2, 4}$
D

${2, 4, 3 \pm 5 i} .$

Solution

Given equation is ${(x - 1)}^{4} + {(x - 5)}^{4} = 82$
Now splitting $- 1$ as $- 3 + 2$ , $- 5$ as $- 2 - 3$

$\Rightarrow {(x - 3 + 2)}^{4} + {(x - 3 - 2)}^{4} = 82$

Let $x - 3 = y$

$\Rightarrow {(y + 2)}^{4} + {(y - 2)}^{4} = 82$

$\Rightarrow [{(y + 2)}^{2} {(y + 2)}^{2}] + [{(y - 2)}^{2} {(y - 2)}^{2}] = 82$

Expand and simplify we get,

$\Rightarrow 2 y^{4} + 48 y^{2} + 32 = 82$

$\Rightarrow 2 (y^{4} + 24 y^{2} + 16) = 82$

$\Rightarrow y^{4} + 24 y^{2} - 25 = 0$

$\Rightarrow y^{2} = - 25, 1$

$\Rightarrow y = \pm 5 i, \pm 1$

$\Rightarrow x - 3 = \pm 5 i, \pm 1$ $(∵ x = y - 3)$

$\Rightarrow x = \pm 5 i + 3, x = \pm 1+3$

$∴$ The solution set of given equation is ${2, 4, 3 \pm 5 i} .$

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