Problem 98 - Introductory and Intermediate Algebra for College Students 4th

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: Write bases in terms of 3

3 is a common base for both sides. Left side is already with base 3. The base on the right side, 9, can also be written as \(3^{2}\). So, \(9^{2x} = (3^{2})^{2x} = 3^{4x}\). So the equation becomes: \(3^{x^2-12} = 3^{4x}\).

Step 2: Equating the exponents

Since the bases are equal, the exponents must also be equal. Thus, set the exponents equal to each other: \(x^2 - 12 = 4x\). This is a quadratic equation.

Step 3: Rearranging the equation

Rearrange the equation to make it easier to solve. It is possible to do this by setting the equation to zero: \(x^{2} - 4x - 12 = 0\)

Step 4: Solving the equation

Now, solve the quadratic equation. This can be done by factoring or applying the quadratic formula, which gives \(x = 6\) or \(x = -2\) as the roots of this equation