SOLUTION: Find all the real numbers that satisfy the given equation x⁴ + (2 - x)⁴ = 34

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Question 1209568: Find all the real numbers that satisfy the given equation
x⁴ + (2 - x)⁴ = 34
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


x%5E4%2B%282-x%29%5E4=34

The expressions "x" and "2-x" are symmetric about the expression "1-x".

So define a new variable:

y = 1-x

Then...

x = 1-y
2-x = 2-(1-y) = 1+y

Now the equation is

%281-y%29%5E4%2B%281%2By%29%5E4=34

%281-4y%2B6y%5E2-4y%5E3%2By%5E4%29%2B%281%2B4y%2B6y%5E2%2B4y%5E3%2By%5E4%29=34

2%2B12y%5E2%2B2y%5E4=34

1%2B6y%5E2%2By%5E4=17

y%5E4%2B6y%5E2-16=0

%28y%5E2%2B8%29%28y%5E2-2%29=0

y%5E2=-8 or y%5E2=2

The problem asks for the real number solutions, so

y%5E2=2
y=0%2B-sqrt%282%29

x=1-y=1%2B-sqrt%282%29

ANSWERS: 1%2Bsqrt%282%29 and 1-sqrt%282%29

Those solutions are easily verified using a graphing calculator like the TI-84.