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Chapter 1: Problem 95

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

Short Answer

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

Step by step solution

01

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.
02

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 \(x^{2} + 2x - 24 = 0\)
03

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 \(x = -6\).
04

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.

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