document.write( "Question 250958: How do I solve (x^2-5x+5)^(x^2-9x+20)=1? \n" ); document.write( "

Algebra.Com's Answer #182965 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
This one is a bit tricky. We have to think out how we can take a number and reise it to a power and get a result of 1. There are three possibilities I can think of:

1 to any power is 1
Any number (excpet zero) to the zero power is 1
-1 to any even power is 1

\n" ); document.write( "So we will look for solutions from all three possibilities.
\n" ); document.write( "1 to any power is 1:
\n" ); document.write( "Solve $\"x%5E2+-5x+%2B+5+=+1\"$
\n" ); document.write( "Subtract 1 from each side:
\n" ); document.write( " $\"x%5E2+-5x+%2B+4+=+0\"$
\n" ); document.write( "Factor:
\n" ); document.write( " $\"x+-4%29%28x-1%29=+0\"$
\n" ); document.write( "Use the Zero Product Property:
\n" ); document.write( " $\"x+-4+=+0\"$ or $\"x+-+1+=+0\"$
\n" ); document.write( "Solve:
\n" ); document.write( " $\"x+=+4\"$ or $\"x+=+1\"$

\n" ); document.write( "Any number but zero to the zero power is 1
\n" ); document.write( "For this we will find the x's that make $\"x%5E2+-9x+%2B20+=+0\"$ and then check to make sure they do not make $\"x%5E2-5x%2B5\"$ zero, too. Solving $\"x%5E2+-9x+%2B20+=+0\"$ ...
\n" ); document.write( "Factor:
\n" ); document.write( " $\"%28x-4%29%28x-5%29+=+0\"$
\n" ); document.write( "Use the Zero Product Property:
\n" ); document.write( " $\"x-4+=+0\"$ or $\"x-5+=+0\"$
\n" ); document.write( "Solve:
\n" ); document.write( " $\"x+=+4\"$ or $\"x+=+5\"$
\n" ); document.write( "Now we check to see if either of these make $\"x%5E2+-+5x+%2B5\"$ zero:
\n" ); document.write( "for x = 4:
\n" ); document.write( " $\"%284%29%5E2+-+5%284%29+%2B+5\"$
\n" ); document.write( " $\"16+-+5%284%29+%2B+5\"$
\n" ); document.write( " $\"16+-+20+%2B+5\"$
\n" ); document.write( " $\"1\"$ This is not 0 so x = 4 is OK.
\n" ); document.write( "for x = 5:
\n" ); document.write( " $\"%285%29%5E2+-+5%285%29+%2B+5\"$
\n" ); document.write( " $\"25+-+5%285%29+%2B+5\"$
\n" ); document.write( " $\"25+-+25+%2B+5\"$
\n" ); document.write( " $\"5\"$ This is not 0 so x = 5 is OK.

\n" ); document.write( "-1 to an even power is 1
\n" ); document.write( "Solve $\"x%5E2+-5x+%2B+5+=+-1\"$ and then see if $\"x%5E2+-9x+%2B+20\"$ is even for those x's.
\n" ); document.write( "Solve $\"x%5E2+-5x+%2B+5+=+-1\"$ :
\n" ); document.write( "Add 1 to each side:
\n" ); document.write( " $\"x%5E2+-5x+%2B+6+=+0\"$
\n" ); document.write( "Factor:
\n" ); document.write( " $\"%28x-3%29%28x-2%29+=+0\"$
\n" ); document.write( "Use the Zero Product Property:
\n" ); document.write( " $\"x=3+=+0\"$ or $\"x-2+=+0\"$
\n" ); document.write( "Solve:
\n" ); document.write( "x = 3 or x = 2
\n" ); document.write( "Now we will see if either of these make the exponent, $\"x%5E2+-9x+%2B+20\"$ , even:
\n" ); document.write( "for x = 3:
\n" ); document.write( " $\"%283%29%5E2+-9%283%29+%2B+20\"$
\n" ); document.write( " $\"9+-9%283%29+%2B+20\"$
\n" ); document.write( " $\"9+-27+%2B+20\"$
\n" ); document.write( " $\"2\"$ This is even so x = 3 is a solution.
\n" ); document.write( "Now we will see if either of these make the exponent, $\"x%5E2+-9x+%2B+20\"$ , even:
\n" ); document.write( "for x = 2:
\n" ); document.write( " $\"%282%29%5E2+-9%282%29+%2B+20\"$
\n" ); document.write( " $\"4+-9%282%29+%2B+20\"$
\n" ); document.write( " $\"4+-18+%2B+20\"$
\n" ); document.write( " $\"2\"$ This is also even so x = 2 is also a solution.

\n" ); document.write( "So the solutions are: x = 4, x = 1, x = 5, x = 3 and x = 2.

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

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