Problem 123 - College Algebra

Question

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

Step-by-Step Solution

Answer

The discriminant provides information about the roots of a quadratic equation. If it's positive, the equation has two distinct real roots. If it's zero, the equation has one real repeated root. If it's negative, the equation has two complex, conjugate roots. The discriminant is calculated as \(b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation.

Step 1: Understanding the Discriminant

The discriminant is part of the quadratic formula, which is utilized to solve quadratic equations. The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The expression under the square root sign, \(b^2 - 4ac\), is the discriminant.

Step 2: Significance of the Discriminant

Depending on the value of the discriminant (whether it's positive, negative, or zero), you can determine the nature of the solutions of the equation. If the discriminant is greater than zero, the equation has two distinct real roots. If it's equal to zero, the equation has exactly one real root (or a repeated root). And if the discriminant is less than zero, the equation has no real roots, but two complex roots instead.

Step 3: Calculating the Discriminant

To find the discriminant of a quadratic equation of the form \(ax^2 + bx + c = 0\), substitute the coefficients \(a\), \(b\), and \(c\) into the formula \(b^2 - 4ac\). Please note that it doesn't provide the solution to the equation itself, but information about the potential solutions.