<PRE><B>Question 1209568</B>
&lt;br&gt;
{{{x^4+(2-x)^4=34}}}&lt;br&gt;
The expressions &quot;x&quot; and &quot;2-x&quot; are symmetric about the expression &quot;1-x&quot;.&lt;br&gt;
So define a new variable:&lt;br&gt;
y = 1-x&lt;br&gt;
Then...&lt;br&gt;
x = 1-y
2-x = 2-(1-y) = 1+y&lt;br&gt;
Now the equation is&lt;br&gt;
{{{(1-y)^4+(1+y)^4=34}}}&lt;br&gt;
{{{(1-4y+6y^2-4y^3+y^4)+(1+4y+6y^2+4y^3+y^4)=34}}}&lt;br&gt;
{{{2+12y^2+2y^4=34}}}&lt;br&gt;
{{{1+6y^2+y^4=17}}}&lt;br&gt;
{{{y^4+6y^2-16=0}}}&lt;br&gt;
{{{(y^2+8)(y^2-2)=0}}}&lt;br&gt;
{{{y^2=-8}}} or {{{y^2=2}}}&lt;br&gt;
The problem asks for the real number solutions, so&lt;br&gt;
{{{y^2=2}}}
{{{y=0+-sqrt(2)}}}&lt;br&gt;
{{{x=1-y=1+-sqrt(2)}}}&lt;br&gt;
ANSWERS: {{{1+sqrt(2)}}} and {{{1-sqrt(2)}}}&lt;br&gt;
Those solutions are easily verified using a graphing calculator like the TI-84.&lt;br&gt;
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