To see how to solve an equation that involves the absolute value of a quadratic polynomial, such as | x² - x | = 6, work Exercises 83–86 in order. For x² - x to have an absolute value equal to 6, what are the two possible values that x may assume? (Hint: One is positive and the other is negative.)

Verified step by step guidance

1. Understand that the equation involves an absolute value: \(|x^2 - x| = 6\). This means the expression inside the absolute value, \(x^2 - x\), can be either 6 or -6.
2. Set up two separate equations to remove the absolute value: \(x^2 - x = 6\) and \(x^2 - x = -6\).
3. Solve the first quadratic equation \(x^2 - x = 6\) by rewriting it as \(x^2 - x - 6 = 0\). Then, factor or use the quadratic formula to find the values of \(x\).
4. Solve the second quadratic equation \(x^2 - x = -6\) by rewriting it as \(x^2 - x + 6 = 0\). Then, factor or use the quadratic formula to find the values of \(x\).
5. Identify the two possible values of \(x\) from the solutions of both equations, noting that one solution will be positive and the other negative as hinted.

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 |A| = B, where B is positive, we set A = B and A = -B, creating two separate equations to solve. This approach helps find all possible values that satisfy the original equation.

Quadratic Polynomials

A quadratic polynomial is a second-degree polynomial of the form ax^2 + bx + c. Understanding its structure is essential for solving equations involving quadratics, as it allows factoring, applying the quadratic formula, or completing the square to find roots or values of x.

Solving Quadratic Equations

Solving quadratic equations involves finding values of x that satisfy the equation, often by factoring, using the quadratic formula, or completing the square. When the quadratic is set equal to a constant (positive or negative), these methods help determine the possible solutions.