Problem 160 - College Algebra

Question

What is the discriminant and what information does it provide about a quadratic equation?

Step-by-Step Solution

Answer

The discriminant is part of the quadratic formula, given by \(b^2 - 4ac\), where 'a', 'b', 'c' are coefficients of the quadratic equation written in standard form \(ax^2 + bx + c = 0\). It provides information about the nature of the roots of the quadratic equation. If the discriminant is greater than zero, the equation has two distinct real roots. If it's equal to zero, there is one real root. If it's less than zero, there are two complex roots.

Step 1: Understanding the Discriminant

The discriminant is given by the formula \(b^2 - 4ac\). It can be used to analyze the roots of a quadratic equation. It does not give the actual roots of the equation, but instead, provides information about the nature of these roots.

Step 2: Identifying the Type of Roots Using Discriminant

If the discriminant is greater than zero (\(b^2 - 4ac > 0\)), the quadratic equation will have two distinct real roots. If the discriminant is equal to zero (\(b^2 - 4ac = 0\)), the equation will have exactly one real root (or a repeated real root). If the discriminant is less than zero (\(b^2 - 4ac < 0\)), the equation will not have any real roots but two complex roots.

Step 3: The Role of Discriminant in Quadratic Formula

The discriminant is a vital component of the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The term \(\sqrt{b^2 - 4ac}\) under the square root sign is the discriminant, and the nature of roots (real or complex) are determined by whether it's positive, zero, or negative respectively.