document.write( "Question 1209568: Find all the real numbers that satisfy the given equation
\n" ); document.write( "x⁴ + (2 - x)⁴ = 34 \n" ); document.write( "

Algebra.Com's Answer #849546 by greenestamps(13200) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( " $\"x%5E4%2B%282-x%29%5E4=34\"$

\n" ); document.write( "The expressions \"x\" and \"2-x\" are symmetric about the expression \"1-x\".

\n" ); document.write( "So define a new variable:

\n" ); document.write( "y = 1-x

\n" ); document.write( "Then...

\n" ); document.write( "x = 1-y
\n" ); document.write( "2-x = 2-(1-y) = 1+y

\n" ); document.write( "Now the equation is

\n" ); document.write( " $\"%281-y%29%5E4%2B%281%2By%29%5E4=34\"$

\n" ); document.write( " $\"%281-4y%2B6y%5E2-4y%5E3%2By%5E4%29%2B%281%2B4y%2B6y%5E2%2B4y%5E3%2By%5E4%29=34\"$

\n" ); document.write( " $\"2%2B12y%5E2%2B2y%5E4=34\"$

\n" ); document.write( " $\"1%2B6y%5E2%2By%5E4=17\"$

\n" ); document.write( " $\"y%5E4%2B6y%5E2-16=0\"$

\n" ); document.write( " $\"%28y%5E2%2B8%29%28y%5E2-2%29=0\"$

\n" ); document.write( " $\"y%5E2=-8\"$ or $\"y%5E2=2\"$

\n" ); document.write( "The problem asks for the real number solutions, so

\n" ); document.write( " $\"y%5E2=2\"$
\n" ); document.write( " $\"y=0%2B-sqrt%282%29\"$

\n" ); document.write( " $\"x=1-y=1%2B-sqrt%282%29\"$

\n" ); document.write( "ANSWERS: $\"1%2Bsqrt%282%29\"$ and $\"1-sqrt%282%29\"$

\n" ); document.write( "Those solutions are easily verified using a graphing calculator like the TI-84.

\n" ); document.write( " \n" ); document.write( "

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