Problem 95 - Algebra and Trigonometry

Question

Solve each equation. $$\left|x^{2}+2 x-36\right|=12$$

Step-by-Step Solution

Answer

The solutions to the equation \(\left|x^{2}+2 x-36\right|=12\) are \(x = -8\) and \(x = 4\).

Step 1: Break Down into Two Equations

To start with, break down the given equation into two separate equations based on the property of the absolute value \nCase 1: \(x^{2} + 2x - 36 = 12\) Case 2: \(x^{2} + 2x - 36 = -12\) \nThis will allow us to solve each case individually.

Step 2: Simplify Each Equations

Simplify each equation to a standard quadratic form by bringing constants to one side of the equation. In both cases, subtract 12 from both sides to simplify each equation. \nFor case 1, this gives \(x^{2} + 2x - 48 = 0\)For case 2, this gives \(x^{2} + 2x - 24 = 0\)

Step 3: Solve Each Equations

Now, solve each equation. The first equation can be factored into \((x - 6)(x + 8) = 0\). Setting each factor equal to zero gives \(x = 6\) and \(x = -8\) . The second equation can be factored into \((x - 4)(x + 6) = 0\). Setting each factor equal to zero gives \(x = 4\) and \(x = -6\).

Step 4: Confirm Solutions

Now, substitute these solutions into the initial equation to confirm if they hold true. Upon evaluating, we find that \(x = -8\) and \(x = 4\) are the valid solutions.