Solve each absolute value equation. |x^2-x-6|=6

Name: Problem 57, Chapter 3: Precalculus Functions and Graphs
Uploaded: 2020-04-22T01:35:40-08:00
Duration: 3 min 19 s
Channel: Shamshad Waris
Description: Solve each absolute value equation. |x^2-x-6|=6

Key Concepts

Factoring

Factoring is a useful technique when handling quadratic equations obtained after splitting an absolute value equation. It involves rewriting the quadratic equation as a product of simpler expressions. By setting each factor equal to zero, one can efficiently find the roots of the equation, making it a valuable tool in algebra.

Absolute Value Equations

An absolute value equation involves the absolute value of an expression being set equal to some number. The key property is that |A| = B (with B nonnegative) has solutions when A = B or A = -B. This means that solving an equation with absolute value typically involves splitting into two separate cases based on the definition of absolute value.

Case Analysis

When solving an absolute value equation, the process of splitting the equation into two cases—one where the inside of the absolute value equals the positive value on the other side, and one where it equals the negative—allows for a systematic method of finding all potential solutions. This technique is fundamental in correctly resolving the equation into simpler forms.

Quadratic Equations

In cases where the expression inside the absolute value is a quadratic function, solving the resulting equations from the case analysis leads to quadratic equations. Solving quadratic equations may involve methods such as factoring, using the quadratic formula, or completing the square, which are foundational techniques in algebra.

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