+349 Find all (real or nonreal) x satisfying
(x - 3)^4 + (x - 5)^4 = -8 + 6(x - 3)(x - 5)^3 - 11(x - 3)^3 (x - 5).
+35 For convienence, why don't we let a = x - 3. Our equation becomes a^4 + (a-2)^4 = -8 + 6(a)(a-2)^3 - 11(a)^3 (a-2). Expanding and simplfiying, we have 7 a^4 + 6 a^3 - 48 a^2 + 16 a + 24= 0. The best way to do this problem now is plug in values using the Rational Root Theorem until you have one, then factor it out.
However, from here, there is no good way to proceed, as the rational root theorem test values all fail. I would like to assume there is a typo with one of the numbers in your equation, because wolframalpha is giving me x≈-0.15270, x≈2.4454, x≈2.4454, x≈4.6898.
Good luck, the general approach should be there, just double check your numbers!
