Problem 97 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 6\) and \(x = -2\).

Step 1: Express both sides with the same base

Rewrite the equation such that both sides have the same base. Since 25 is a power of 5, rewrite \(25 = 5^{2}\). So, the equation becomes \(5^{x^{2}-12} = (5^{2})^{2x}\). Now simplify the right side using the property of exponentiation \(a^{m \cdot n} = a^{m \cdot n}\), to obtain \(5^{x^{2}-12} = 5^{4x}\).

Step 2: Make the exponents equal

Since now both sides of the equation share the same base, equate their exponents. This gives you a simple algebraic equation: \(x^{2}-12 = 4x\).

Step 3: Re-arrange and solve quadratic equation

Re-arrange the equation from step 2 to have it in the standard quadratic equation format. We get: \(x^{2} - 4x -12 = 0\). Now factorize and solve for \(x\): \((x - 6)(x + 2) = 0\). Solving this gives two possible solutions \(x = 6\) and \(x = -2\).