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Chapter 4: Problem 101

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

Short Answer

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

Step by step solution

01

Make the Bases the Same

Rewrite 25 as \(5^2\). This gives the equation: \(5^{x^{2}-12} = (5^2)^{2x}\).
02

Simplify the Equation

Now we simplify the equation to have a single power of 5 on each side. This gives: \(5^{x^{2}-12} = 5^{4x}\).
03

Set the Exponents Equal to Each Other

If the bases are equal, then the exponents must also be equal. So, we set them equal: \(x^2 - 12 = 4x\).
04

Solve the Quadratic Equation

Rearrange the terms and simplify into a standard quadratic form to find the roots. The quadratic equation becomes: \(x^2 - 4x - 12 = 0\). Factoring the equation yields \((x-6)(x+2) = 0\).
05

Solve for x

Setting each factor equal to zero gives the solutions: \(x = 6\) or \(x = -2\).

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