Problem 101 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The solutions are \( x = 6\) and \(x = -2\)

Step 1: Express both sides with the same base

Since both 5 and 25 are powers of 5, express 25 as \(5^{2}\). Then, the equation becomes: \[5^{x^{2}-12}= (5^{2})^{2x}\] or \[5^{x^{2}-12} = 5^{4x} \]

Step 2: Set the exponents equal to each other

Now that both sides have the same base, you can set the exponents equal to each other: \[ x^{2}-12 = 4x \]

Step 3: Rearrange the equation to Quadratic Form

Rearrange the equation into standard quadratic form: \[x^2 - 4x - 12 = 0\]

Step 4: Solve the quadratic equation

Next, solve the quadratic equation, by either factoring, using quadratic formula, or completing the square. Here, we can use factoring to get \((x - 6) (x + 2) = 0\) which yields \(x = 6\) and \(x = -2\)