Let's solve the equation step-by-step.
Given:
[tex]\[
2^{x^2 - 53x + 25} = 32^{2 - 9x}
\][/tex]
First, recognize that [tex]\(32\)[/tex] can be written in terms of a base of [tex]\(2\)[/tex] since [tex]\(32 = 2^5\)[/tex]. Thus, the equation can be rewritten:
[tex]\[
2^{x^2 - 53x + 25} = (2^5)^{2 - 9x}
\][/tex]
Next, apply the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
2^{x^2 - 53x + 25} = 2^{5(2 - 9x)}
\][/tex]
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
x^2 - 53x + 25 = 5(2 - 9x)
\][/tex]
Expand the right-hand side:
[tex]\[
x^2 - 53x + 25 = 10 - 45x
\][/tex]
Move all terms to one side of the equation to set it to zero:
[tex]\[
x^2 - 53x + 25 - 10 + 45x = 0
\][/tex]
Simplify the equation:
[tex]\[
x^2 - 8x + 15 = 0
\][/tex]
This is a quadratic equation. Use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a=1\)[/tex], [tex]\(b=-8\)[/tex], and [tex]\(c=15\)[/tex]:
[tex]\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 15}}{2 \cdot 1}
\][/tex]
Calculate the discriminant:
[tex]\[
x = \frac{8 \pm \sqrt{64 - 60}}{2}
\][/tex]
[tex]\[
x = \frac{8 \pm \sqrt{4}}{2}
\][/tex]
[tex]\[
x = \frac{8 \pm 2}{2}
\][/tex]
This gives us two solutions:
[tex]\[
x = \frac{8 + 2}{2} = 5 \quad \text{and} \quad x = \frac{8 - 2}{2} = 3
\][/tex]
Thus, the solutions to the equation are:
[tex]\[
\boxed{3 \text{ and } 5}
\][/tex]
