SOLUTION: How do I solve (x^2-5x+5)^(x^2-9x+20)=1?

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Question 250958: How do I solve (x^2-5x+5)^(x^2-9x+20)=1?
Answer by jsmallt9(3758)   (Show Source): 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:
So we will look for solutions from all three possibilities.
1 to any power is 1:
Solve
Subtract 1 from each side:

Factor:

Use the Zero Product Property:
or
Solve:
or

Any number but zero to the zero power is 1
For this we will find the x's that make and then check to make sure they do not make zero, too. Solving ...
Factor:

Use the Zero Product Property:
or
Solve:
or
Now we check to see if either of these make zero:
for x = 4:



This is not 0 so x = 4 is OK.
for x = 5:



This is not 0 so x = 5 is OK.

-1 to an even power is 1
Solve and then see if is even for those x's.
Solve :
Add 1 to each side:

Factor:

Use the Zero Product Property:
or
Solve:
x = 3 or x = 2
Now we will see if either of these make the exponent, , even:
for x = 3:



This is even so x = 3 is a solution.
Now we will see if either of these make the exponent, , even:
for x = 2:



This is also even so x = 2 is also a solution.

So the solutions are: x = 4, x = 1, x = 5, x = 3 and x = 2.
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