When the discriminant is a positive number but not a perfect square, what type of roots can be expected?

When the discriminant of a quadratic equation is a positive number but not a perfect square, it indicates that the equation has two distinct real roots. The distinction that the number is not a perfect square means that these roots will be irrational.

The discriminant, given by the expression ( b^2 - 4ac ) for a quadratic equation of the form ( ax^2 + bx + c = 0 ), serves to determine the nature of the roots. A positive discriminant signals that the parabola intersects the x-axis at two points, hence there are two roots. However, when the discriminant is not a perfect square, the solutions cannot be expressed as simple fractions (or whole numbers), thus resulting in two real irrational roots.

This reinforces the understanding of how the discriminant influences the nature of the roots in quadratic equations, helping students identify the outcomes based on the value of the discriminant in various scenarios.