To solve the equation [tex]\(5^{x^2 + 7x} = 25^{7x - 6}\)[/tex], follow these steps:
1. Express 25 as a power of 5:
[tex]\[
25 = 5^2
\][/tex]
2. Rewrite the given equation using this identity:
[tex]\[
5^{x^2 + 7x} = (5^2)^{7x - 6}
\][/tex]
3. Simplify the right-hand side:
[tex]\[
(5^2)^{7x - 6} = 5^{2(7x - 6)} = 5^{14x - 12}
\][/tex]
4. Since the bases are the same, set the exponents equal to each other:
[tex]\[
x^2 + 7x = 14x - 12
\][/tex]
5. Rearrange the equation to set it to 0:
[tex]\[
x^2 + 7x - 14x + 12 = 0 \implies x^2 - 7x + 12 = 0
\][/tex]
6. Solve the quadratic equation [tex]\(x^2 - 7x + 12 = 0\)[/tex] using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = -7\)[/tex], and [tex]\(c = 12\)[/tex].
7. Calculate the discriminant:
[tex]\[
b^2 - 4ac = (-7)^2 - 4(1)(12) = 49 - 48 = 1
\][/tex]
8. Find the square root of the discriminant:
[tex]\[
\sqrt{1} = 1
\][/tex]
9. Find the solutions:
[tex]\[
x = \frac{-(-7) \pm 1}{2(1)} = \frac{7 \pm 1}{2}
\][/tex]
This yields two solutions:
[tex]\[
x_1 = \frac{7 + 1}{2} = 4
\][/tex]
[tex]\[
x_2 = \frac{7 - 1}{2} = 3
\][/tex]
Therefore, the solutions are:
[tex]\[
x = 4, 3
\][/tex]
