Problem 102 - Algebra and Trigonometry

Question

Solve each equation. $$ 3^{x^{2}-12}=9^{2 x} $$

Step-by-Step Solution

Answer

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

Step 1: Write both sides of the equation with the same base

Manipulate the equation so that both sides are of the same base. Since 9 is 3 squared, replace 9 with \(3^{2}\) in the equation. So, the equation becomes \(3^{x^{2} - 12} = (3^{2})^{2x}\]

Step 2: Simplify the equation

By the rules of exponents, you can multiply the exponents when they're inside brackets. Therefore, the equation can be rewritten as \(3^{x^{2} -12} = 3^{4x}\)

Step 3: Equate the Exponents

With the same bases on either side of the equation, you can simply equate the exponents to each other, giving us \(x^{2} - 12 = 4x\)

Step 4: Rearrange the resulting equation

The resulting equation \(x^{2} - 12 = 4x\) is a quadratic equation. Rearrange the terms to bring them all to one side to get \(x^{2} - 4x - 12 = 0\)

Step 5: Solve the quadratic equation

Solve the quadratic equation \(x^{2} - 4x - 12 = 0\) using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\), or factoring if possible. When factoring the quadratic equation, it becomes \((x - 6)(x + 2) = 0\), and we find that \(x = 6\) or \(x = -2\) - both are the solutions to the equation.