In quadratic equations, what does the term "discriminant" determine?

The discriminant is a crucial component of the quadratic formula, which is used to find the solutions of a quadratic equation in the standard form ( ax^2 + bx + c = 0 ). The discriminant is given by the expression ( D = b^2 - 4ac ).

The significance of the discriminant lies in its ability to determine the nature and number of solutions to the quadratic equation:

• If ( D > 0 ), there are two distinct real solutions, meaning the parabola intersects the x-axis at two points.
• If ( D = 0 ), there is exactly one real solution (also called a repeated or double root), indicating that the vertex of the parabola touches the x-axis.
• If ( D < 0 ), there are no real solutions, only complex solutions, meaning the parabola does not intersect the x-axis at all.

Thus, the correct interpretation is that the discriminant effectively tells us how many solutions a quadratic equation has, which is why it is correct to state that it determines the number of solutions.