To see how to solve an equation that involves the absolute value of a quadratic polynomial, such as | x^2 - x | = 6, work Exercises 83–86 in order. For x^2 - 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.)

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. For example, |x| = x if x is positive, and |x| = -x if x is negative. In the context of equations, the absolute value can create two scenarios: one where the expression inside is equal to the positive value and another where it is equal to the negative value.

Quadratic Polynomials

A quadratic polynomial is a polynomial of degree two, typically expressed in the form ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic polynomial is a parabola, which can open upwards or downwards depending on the sign of 'a'. Understanding the properties of quadratics is essential for solving equations involving them, especially when combined with absolute values.

Solving Absolute Value Equations

To solve an equation involving absolute values, such as |f(x)| = k, where k is a positive number, you must set up two separate equations: f(x) = k and f(x) = -k. This approach allows you to find all possible solutions for x. In the given problem, you would set up the equations x^2 - x = 6 and x^2 - x = -6 to find the values of x that satisfy the original absolute value equation.