In Exercises 101–106, solve each equation. |x^2 + 2x - 36| = 12

Verified step by step guidance

1. Identify that the equation involves an absolute value: |x^2 + 2x - 36| = 12.
2. Recognize that the absolute value equation |A| = B can be split into two separate equations: A = B and A = -B.
3. Set up the first equation: x^2 + 2x - 36 = 12.
4. Set up the second equation: x^2 + 2x - 36 = -12.
5. Solve each quadratic equation separately by moving all terms to one side to form standard quadratic equations and then use the quadratic formula or factoring to find the solutions for x.

Key Concepts

Absolute Value

Absolute value measures the distance of a number from zero on the number line, regardless of direction. For any real number x, the absolute value is defined as |x| = x if x ≥ 0 and |x| = -x if x < 0. In the context of equations, it creates two separate cases to consider, as the expression inside the absolute value can be either positive or negative.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to quadratic equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding how to manipulate and solve these equations is crucial for solving problems involving absolute values.

Case Analysis

Case analysis is a problem-solving technique used to break down complex problems into simpler, manageable parts. In the context of absolute value equations, it involves creating separate equations for each possible case (positive and negative) derived from the absolute value expression. This method allows for a systematic approach to finding all possible solutions to the original equation.