SOLUTION: 9^x-2(3^x+2)+81=0

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Question 1176692: 9^x-2(3^x+2)+81=0
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

We'll apply the rule
%28a%5Eb%29%5Ec+=+a%5E%28b%2Ac%29
to say that:
9%5Ex+=+%283%5E2%29%5Ex

9%5Ex+=+3%5E%282x%29

9%5Ex+=+%283%5Ex%29%5E2

The equation
9%5Ex-2%283%5Ex%2B2%29%2B81=0
then becomes
%283%5Ex%29%5E2-2%283%5Ex%2B2%29%2B81=0

Let w+=+3%5Ex
Replace every copy of 3%5Ex with w

We go from
%283%5Ex%29%5E2-2%283%5Ex%2B2%29%2B81=0
to
w%5E2-2%28w%2B2%29%2B81=0

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Let's solve for w

w%5E2-2%28w%2B2%29%2B81=0

w%5E2-2w-4%2B81=0

w%5E2-2w%2B77=0

Apply the quadratic formula
w+=+%28-b%2Bsqrt%28b%5E2-4ac%29%29%2F%282a%29 or w+=+%28-b-sqrt%28b%5E2-4ac%29%29%2F%282a%29

w+=+%28-%28-2%29%2Bsqrt%28%28-2%29%5E2-4%281%29%2877%29%29%29%2F%282%281%29%29 or w+=+%28-%28-2%29-sqrt%28%28-2%29%5E2-4%281%29%2877%29%29%29%2F%282%281%29%29

w+=+%284%2Bsqrt%28-304%29%29%2F%282%29 or w+=+%284-sqrt%28-304%29%29%2F%282%29

We run into a problem. The negative under the square root means we'll have complex numbers. I'll assume your teacher hasn't covered the concept yet (please let me know if otherwise)

So because w%5E2-2w%2B77=0 has complex solutions, then so does the original equation 9%5Ex-2%283%5Ex%2B2%29%2B81=0


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Now that I think about it further, it's possible that the equation you were given is 9%5Ex-2%2A3%5E%28x%2B2%29%2B81=0 instead of 9%5Ex-2%2A%283%5E%28x%29%2B2%29%2B81=0. Note how the 2 has moved from being in the exponent and then not in the exponent.

If the 2 is in the exponent, then,
9%5Ex-2%2A3%5E%28x%2B2%29%2B81=0

%283%5Ex%29%5E2-2%2A%283%5Ex%29%2A3%5E%282%29%2B81=0

%283%5Ex%29%5E2-2%2A%283%5Ex%29%2A9%2B81=0

w%5E2-18w%2B81=0

%28w-9%29%5E2=0

w-9+=+sqrt%280%29

w-9=0

w=9

If w = 9, then,
w+=+3%5Ex

9+=+3%5Ex

3%5Ex+=+9

3%5Ex+=+3%5E2

x+=+2

So once again, if the equation you were given was 9%5Ex-2%2A3%5E%28x%2B2%29%2B81=0 (note the 2 in the exponent), then it leads to the single solution of x+=+2

We can check this by plugging x = 2 back into the original equation to get...
9%5Ex-2%2A3%5E%28x%2B2%29%2B81=0

9%5E2-2%2A3%5E%282%2B2%29%2B81=0

9%5E2-2%2A3%5E4%2B81=0

81-2%2A81%2B81=0

81-162%2B81=0

-81%2B81=0

0=0
We get the same thing on both sides, so the answer is confirmed for that equation.