Put y=(x−1)+(x−5)2=x−3x=y+3 The given equation reduces to(y+2)4+(y−2)4=82y4+24y2−25=0y2+25y2−1=0y=5i, ±1x−3=5i, ±1x=3+5i, 3±1x=3+5i,4,2

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$(x - 1)^{4} + (x - 5)^{4} = 82$

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x=1

x=4

x=2

$x = 3 \pm 5 i, 4, 2$

answer is D.

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Detailed Solution

Put $y = \frac{(x - 1) + (x - 5)}{2} = x - 3$

x=y+3 The given equation reduces to

$\begin{array}{l} (y + 2)^{4} + (y - 2)^{4} = 82 \\ y^{4} + 24 y^{2} - 25 = 0 \\ (y^{2} + 25) (y^{2} - 1) = 0 \\ y = 5 i, \pm 1 \\ x - 3 = 5 i, \pm 1 \\ x = 3 + 5 i, 3 \pm 1 \\ x = 3 + 5 i, 4, 2 \end{array}$

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