Solve $x^4+(4-x)^4=32$ Equation

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SUMMARY

The equation $x^4+(4-x)^4=32$ can be solved using the substitution $t=x-2$. This transformation simplifies the equation, allowing for easier manipulation and solution. The discussion highlights the effectiveness of this substitution method in tackling polynomial equations of this nature.

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solakis1
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solve the following equation:

$x^4+(4-x)^4=32$
 
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My attempt.
Looks like symmetry would be helpful.
Let's set $x=y+2$.
Then we get
$$(y+2)^4 + (4-(y+2))^4=(y+2)^4 + (y-2)^4=2y^4+48y^2+32=32 \implies y^2(y^2+24)=0\implies y=0 \implies x=2$$
 
solakis said:
solve the following equation:

$x^4+(4-x)^4=32$

Do an average substitution $t=x-2$. I have a solution to a similar equation:

 

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