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

Verified step by step guidance

1. Recognize that the equation involves an absolute value: \(|x^2 + 2x - 36| = 12\). This means the expression inside the absolute value can be either 12 or -12.
2. Set up two separate equations to remove the absolute value: 1) \(x^2 + 2x - 36 = 12\) 2) \(x^2 + 2x - 36 = -12\)
3. Solve the first quadratic equation: \(x^2 + 2x - 36 = 12\). Start by moving all terms to one side to set the equation to zero: \(x^2 + 2x - 36 - 12 = 0\), which simplifies to \(x^2 + 2x - 48 = 0\).
4. Solve the second quadratic equation: \(x^2 + 2x - 36 = -12\). Move all terms to one side: \(x^2 + 2x - 36 + 12 = 0\), which simplifies to \(x^2 + 2x - 24 = 0\).
5. For each quadratic equation, use factoring, completing the square, or the quadratic formula to find the values of \(x\) that satisfy the equation.

Key Concepts

Absolute Value Equations

An absolute value equation involves expressions within absolute value bars, which represent the distance from zero on the number line. To solve, set the expression inside equal to both the positive and negative values of the number on the other side, since |A| = B implies A = B or A = -B.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form ax² + bx + c = 0. They can be solved by factoring, completing the square, or using the quadratic formula. Solutions may be real or complex numbers.

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of two binomials. This method is useful for solving quadratic equations by setting each factor equal to zero. Recognizing factorable forms simplifies finding the roots of the equation.