I am trying to write a mathematical paper and I am including the following paragraph about the discriminant of the quadratic equation. I am trying to wrap my head around this concept and I wanted to make sure my logic was correct.
The discriminant reveals the nature of the roots of a quadratic equation given that $a, b$ and $c$ are rational numbers. When calculated it determines the number of real roots, or in other words, the number of x-intercepts, associated with a quadratic equation. For example, if $b^2 - 4ac = 0$, then the quadratic equation will have exactly one real number solution (with duplicity.) In other words, there is one solution that is repeated resulting in the curve intersecting the $x$-axis at one point. If $b^2 - 4ac > 0$ (in other words the discriminant is a positive number), then the quadratic equation will have exactly two real number solutions resulting in the curve intersecting the $x$-axis at two distinct points. If $b^2 - 4ac < 0$ (in other words the discriminant is a negative number), then the quadratic equation will have no real solutions. Instead there will be two complex number solutions resulting in the curve having no intersection points on the $x$-axis.