Problem 102 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: Express 9 as a power of 3

Rewrite \(9^{2x}\) as \((3^2)^{2x}\) which simplifies to \(3^{4x}\). Now the equation can be rewritten as \(3^{x^2 - 12} = 3^{4x}\).

Step 2: Equate the exponents

Since the bases are the same (3 in both instances), the exponents must be equal. \[x^2 - 12 = 4x\]

Step 3: Solve the quadratic equation

Rearrange the equation \(x^2 - 4x - 12 = 0\), which is a quadratic equation in standard form \[ ax^2 + bx + c = 0. \] Next find the roots using the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where a=1, b=-4, and c=-12. Substituting the given values gives the solution \[ x = \frac{4 \pm \sqrt{(( -4 )^2 - 4( 1)( -12 ))}}{2*1} \] This simplifies to \[ x = \frac{4 \pm \sqrt{64}}{2} \] which further simplifies to \[ x = 2 \pm \sqrt{16} \] yielding the two roots \[ x = 2 + 4 = 6 \] and \[ x = 2 - 4 = -2 \]