Problem 160 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The discriminant is a component of the quadratic formula that determines the nature of the roots of the equation. Specifically, if the discriminant is greater than 0, the equation has two distinct real roots; if it's equal to 0, the equation has two equal real roots (or one real root); and if it's less than 0, the equation has no real roots but two complex roots.

Step 1: Find the discriminant

Substitute the values of 'a', 'b', and 'c' from the quadratic equation into the formula for the discriminant: \( b^2 - 4ac \). The result obtained is called the discriminant and denoted as 'D'. If the equation is \( ax^2 + bx + c = 0 \), then \( D = b^2 - 4ac \).

Step 2: Determine the nature of roots

Use the value of 'D' to determine the nature of the roots of the quadratic equation:1. If \( D > 0 \), the equation has two distinct real roots.2. If \( D = 0 \), the equation has two equal real roots (or one real root).3. If \( D < 0 \), the equation has no real roots. Instead, the equation has two complex roots.