Solve each equation. $5^{x^2 - 12} = 25^{2x}$

Verified step by step guidance

1. Recognize that both sides of the equation involve exponential expressions with base 5 or powers of 5. Rewrite 25 as a power of 5: since \(25 = 5^2\), rewrite the right side as \(25^{2x} = (5^2)^{2x}\).
2. Apply the power of a power property: \((a^m)^n = a^{mn}\). So, \((5^2)^{2x} = 5^{4x}\). Now the equation becomes \(5^{x^2 - 12} = 5^{4x}\).
3. Since the bases are the same and the expressions are equal, set the exponents equal to each other: \(x^2 - 12 = 4x\).
4. Rewrite the equation to standard quadratic form by moving all terms to one side: \(x^2 - 4x - 12 = 0\).
5. Solve the quadratic equation \(x^2 - 4x - 12 = 0\) using factoring, completing the square, or the quadratic formula to find the values of \(x\).

Key Concepts

Properties of Exponents

Understanding how to manipulate expressions with exponents is essential. This includes knowing that a^(m*n) = (a^m)^n and that a^m * a^n = a^(m+n). These properties allow rewriting and simplifying exponential equations to make them easier to solve.

Expressing Numbers with the Same Base

To solve exponential equations, it helps to rewrite both sides with the same base. For example, 25 can be expressed as 5^2. This allows setting the exponents equal to each other when the bases are the same, simplifying the equation to an algebraic form.

Solving Quadratic Equations

After equating exponents, the resulting equation may be quadratic in form. Knowing how to solve quadratic equations using factoring, completing the square, or the quadratic formula is necessary to find the values of the variable.