The simple rules about real roots of polynomials with real coefficients of degree n are:

(1) The number of real roots cannot exceed n;

(2) If n is odd, there is at least one real root; and

(3) If n is even, there may be no real roots.

If we apply those rules to a quadratic, a polynomial of degree 2, we find that it may have no real root, 1 real root, or 2 **distinct** real roots. How do you know which of those situations apply to a specific case?

You look at the discriminant of the quadratic.

\(\displaystyle b^2 - 4ac < 0 \implies 0 \text { real roots;}\)

\(\displaystyle b^2 - 4ac = 0 \implies 1 \text { real root; and}\)

\(\displaystyle b^2 - 4ac > 0 \implies 2 \text { distinct real roots.}\)

For reasons that make sense, some people prefer to say that a quadratic always has two roots, but it may have no real roots, and if it has real roots, they may be duplicates. I prefer my way because it relates the three possible signs of the discriminant to three possible numbers of distinct, real roots. Just remember whichever seems more intuitive to you.

In any case, good job.