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Question

Solve (x - 3)^4 + (x - 5)^4 = -8.Enter all the solutions, separated by commas.

Accepted Answer

We can introduce symmetry into the equation by letting

z = x - 4.

Then

x = z + 4,

so the equation becomes

(z + 1)^{4} + (z - 1)^{4} = - 8.

This simplifies to

2 z^{4} + 12 z^{2} + 10 = 0,

or

z^{4} + 6 z^{2} + 5 = 0.

This factors as

(z^{2} + 1) (z^{2} + 5) = 0,

so

z = \pm i

or

z = \pm i 5 .

Therefore, the solutions are

4 + i, 4 - i, 4 + i 5, 4 - i 5 .