Answer:
x = 3/2
Explanation:
I like to write the equation in the form f(x) = 0 and use a graphing calculator to solve it. Here, that can be achieved by subtracting the right side from both sides:
9^x -2^(x +1/2) -2^(x +7/2) +3^(2x -1) = 0
See the attachment for the graphical solution.
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This sort of equation is solved by rearranging it to the form ...
a^x = b
Then the solution is ...
x = log(b)/log(a)
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Separating powers of 2 and powers of 3, we get ...
9^x +3^(2x-1) = 2^(x +7/2) +2^(x +1/2)
3^(2x) +3^(2x)·3^(-1) = 2^x(2^(7/2) +2^(1/2)) . . . . use 9=3^2; factor out 2^x
3^(2x)·(4/3) = 2^x(9√2) . . . . . simplify
(9/2)^x = (9√2)·(3/4) . . . . . . . multiply by (3/4)2^-x
Taking the log of both sides, and using 9=3^2, we get ...
x·(2log(3) -log(2)) = (2log(3) +(1/2)log(2) +log(3) -2log(2) = 3log(3) -(3/2)log(2)
Dividing by the coefficient of x gives ...
x = (3log(3) -(3/2)log(2))/(2log(3) -log(2))
x = 3/2