What is the situation when the discriminant is a positive perfect square?

When the discriminant of a quadratic equation is a positive perfect square, it indicates the nature of the roots of the equation. The discriminant, given by the formula (b^2 - 4ac), provides insight into the types of solutions:

1. A positive discriminant means that there are two distinct real roots.

2. If this positive discriminant is also a perfect square, it suggests that these roots can be expressed as rational numbers. This is because the square root of a perfect square is an integer; thus, it does not introduce any irrational components in the solutions.

For instance, consider the quadratic equation formed by the discriminant (9) (which is (3^2)). The roots derived from this would take the form of rational numbers, as (\frac{-b \pm \sqrt{9}}{2a}) results in rational outcomes.

This leads to the conclusion that when the discriminant is a positive perfect square, the quadratic equation will yield two real rational roots.