Lesson 3: The nature of the roots

From previous discussions, we know that an equation of the form $y=a{x}^2+bx+c$  has two roots, but the roots are not always distinct. Take, for example, the equation

We can see that x = –2 when y = 0. This occurs twice, so we call it a double root. It is a single distinct result, but still considered as two roots.

What happens when the equation will not factor over the set of real numbers? Consider

From equation 1.1 it is clear that when $y=0,x=\pm \sqrt{-9}=\pm 3i$ . In this situation, we have two distinct roots, both of which are complex numbers. It is possible to determine the "nature of the roots" of a quadratic equation without completely solving the equation.

Determining the "nature of the roots" requires answering the question, "Does the equation have two real roots, a double root, or two complex roots?" Recall the quadratic formula, useful for solving equations of the form

The solution to this equation will produce two distinct real roots when the radicand is positive, a single result (indicating a double root) when the radicand is zero, and two distinct complex conjugate roots when the radicand is negative. The radicand in the quadratic formula is called the discriminant.