# Solve the equation.

Solve the equation
$\left&space;(&space;x+1&space;\right&space;)^{4}+x^{4}=17$

Write  x+1=t

$t^{4}+\left&space;(&space;t-1&space;\right&space;)^{4}=17$

$t^{4}+\left&space;(&space;t-1&space;\right&space;)^{4}=16+1$

$t^{4}-1+\left&space;(&space;t-1&space;\right&space;)^{4}=16$

$\left&space;(&space;t^{2}-1&space;\right&space;)\left&space;(&space;t^{2}+1&space;\right&space;)+\left&space;(&space;t-1&space;\right&space;)^{4}=16$

$\left&space;(&space;t-1&space;\right&space;)\left&space;(&space;t+1&space;\right&space;)\left&space;(&space;t^{2}+1&space;\right&space;)+\left&space;(&space;t-1&space;\right&space;)\left&space;(&space;t-1&space;\right&space;)^{3}=16$

$\left&space;(&space;t-1&space;\right&space;)\left&space;[&space;\left&space;(&space;t+1&space;\right&space;)\left&space;(&space;t^{2}+1&space;\right&space;)+\left&space;(&space;t-1&space;\right&space;)^{3}&space;\right&space;]=16$

$\left&space;(&space;t-1&space;\right&space;)\left&space;(&space;t^{3}+t^{2}+t+1+t^{3}-3t^{2}+3t-1&space;\right&space;)=16$

$\left&space;(&space;t-1&space;\right&space;)\left&space;(&space;2t^{3}-2t^{2}+4t&space;\right&space;)=16$

$2t\left&space;(&space;t-1&space;\right&space;)\left&space;(&space;t^{2}-t+2&space;\right&space;)=16$

$t\left&space;(&space;t-1&space;\right&space;)\left&space;(&space;t^{2}-t+2&space;\right&space;)=8$

$\left&space;(&space;t^{2}-t&space;\right&space;)\left&space;(&space;t^{2}-t+2&space;\right&space;)=8$

Plug $t^{2}-t=a$

$a\left&space;(&space;a+2&space;\right&space;)=8$

$a^{2}+2a-8=0$

$D=36,&space;a_{1}=-4\wedge&space;a_{2}=2$

Go back to the replacement of  t

$t^{2}-t=-4\Leftrightarrow&space;t^{2}-t+4=0$ no real solution D<0

OSE

$t^{2}-t=2\Leftrightarrow&space;t^{2}-t-2=0$ has two solutions -1 and  2.

$t=-1\Rightarrow&space;x+1=-1\Rightarrow&space;x=-2$

$t=2\Rightarrow&space;x+1=2\Rightarrow&space;x=1$

The Solutions of the equations are $A=\left&space;\{&space;-2;1&space;\right&space;\}$.