To solve the equation
[tex]\[
9^{2x - 5} = 3^{x^2 + 11x - 18},
\][/tex]
we'll start by expressing the equation with the same base. Notice that [tex]\( 9 \)[/tex] can be written as [tex]\( 3^2 \)[/tex].
First, rewrite [tex]\( 9 \)[/tex] as [tex]\( 3^2 \)[/tex]:
[tex]\[
(3^2)^{2x - 5} = 3^{x^2 + 11x - 18}
\][/tex]
Next, simplify the left side of the equation:
[tex]\[
(3^2)^{2x - 5} = 3^{(2)(2x - 5)} = 3^{4x - 10}
\][/tex]
So the equation becomes:
[tex]\[
3^{4x - 10} = 3^{x^2 + 11x - 18}
\][/tex]
Since the bases are now the same, we can set the exponents equal to each other:
[tex]\[
4x - 10 = x^2 + 11x - 18
\][/tex]
Move all the terms to one side of the equation to set it to zero:
[tex]\[
0 = x^2 + 11x - 18 - 4x + 10
\][/tex]
Simplify the equation:
[tex]\[
0 = x^2 + 7x - 8
\][/tex]
We solve this quadratic equation by factoring or using the quadratic formula. Factoring gives:
[tex]\[
x^2 + 7x - 8 = (x + 8)(x - 1)
\][/tex]
Setting each factor to zero:
[tex]\[
x + 8 = 0 \quad \text{or} \quad x - 1 = 0
\][/tex]
Solving these equations, we find:
[tex]\[
x = -8 \quad \text{or} \quad x = 1
\][/tex]
Therefore, the solutions to the equation [tex]\( 9^{2x - 5} = 3^{x^2 + 11x - 18} \)[/tex] are:
[tex]\[
x = -8, \quad x = 1
\][/tex]
