What does the term 'discriminant' refer to in a quadratic equation?

The term 'discriminant' in the context of a quadratic equation is defined as the expression found under the square root in the quadratic formula, given by ( b^2 - 4ac ), where ( a ), ( b ), and ( c ) are the coefficients of the quadratic equation in the standard form ( ax^2 + bx + c = 0 ).

The value of the discriminant is critical because it provides key information about the nature of the roots of the quadratic equation. Specifically:

1. If the discriminant is greater than zero, the equation has two distinct real roots. This means the parabola intersects the x-axis at two points.

2. If the discriminant is equal to zero, the equation has exactly one real root, or a repeated root. In this case, the vertex of the parabola touches the x-axis, creating a situation where the graph is tangent to the axis.

3. If the discriminant is less than zero, the equation has no real roots, indicating that the graph of the parabola does not intersect the x-axis at all; instead, it lies entirely above or below the x-axis.

Given this understanding, identifying the discriminant as the quantity that determines the number