FORUM REVIVE(hopefully)

Here's a little fun substitution problem!
Find all real numbers x such that $$(x^2  x  2)^4 + (2x+1)^4 = (x^2+x1)^4.$$

The answers are 1, 0.5, and 2.

@excitedarmadillo0 Correct! Now, what was your solution to this?

Painfully expand and simplify.

@excitedarmadillo0 uhh :concern: are you ok?
ANYWAYS, here's the noniwanttodieandi'mamasochist solution!
$$a = x^2  x  2, b = 2x+1. $$
Let us assume thatThus, the equation can be written as $$a^4 + b^4 = (a+b)^4 = a^4 + 4a^3b+6a^2b^2+4ab^3+b^4.$$
Simplifying, we get $$2ab(2a^2 + 3ab+2b^2) = 0.$$
Assuming that a=0, we get that $$x^2x2 = 0,$$ thus x = 1 or x = 2, which are the solutions of the equation.
Assume now that b=0, we get that $$x = \frac{1}{2}.$$
Finally, assume that ab does not equal 0, then we get $$2a^2+3ab+2b^2 = 0.$$ As this is a quadratic equation in a, the discriminant is $$7*b^2<0.$$
Thus, this equation has no solution, meaning that we have 3 solutions: $$1, \frac{1}{2}, 2 \blacksquare$$

Yes very fun and totally not pain and suffering.