FORUM REVIVE(hopefully)
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Here's a little fun substitution problem!
Find all real numbers x such that $$(x^2 - x - 2)^4 + (2x+1)^4 = (x^2+x-1)^4.$$
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The answers are -1, -0.5, and 2.
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@excitedarmadillo-0 Correct! Now, what was your solution to this?
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Painfully expand and simplify.
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@excitedarmadillo-0 uhh :concern: are you ok?
ANYWAYS, here's the non-i-want-to-die-and-i'm-a-masochist 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 $$x^2-x-2 = 0,$$ 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: $$-1, -\frac{1}{2}, 2 \blacksquare$$
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Yes very fun and totally not pain and suffering.
