What value does a discriminant of zero indicate?

A discriminant of zero indicates that a quadratic equation has exactly one real root, also known as a double root. This situation arises because when the discriminant, which is calculated as ( b^2 - 4ac ), equals zero, it means the quadratic formula results in one unique solution rather than two distinct solutions.

To elaborate, the quadratic formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) relies on the discriminant for determining the nature of the roots. When the discriminant is zero, the square root term becomes zero, simplifying the formula to ( x = \frac{-b}{2a} ). This shows that the graph of the quadratic function touches the x-axis at a single point, leading to the conclusion that there is only one real root.