In the Quadratic Formula, what does it mean if $b^2-4ac>0$, $b^2-4ac<0$, and $b^2-4ac=0$?

Question

Concerning the Quadratic Formula:

What does it mean if $b^2-4ac>0$, $b^2-4ac<0$, and $b^2-4ac=0$?

Answer 1

$\Delta={ b^2-4ac}>0$ means that the equation has two real solutions.

$\Delta= { b^2-4ac}<0$ means that the equation has no real solutions, but two complex solutions.

$\Delta={ b^2-4ac}=0$ means that the equation has one solution.

Answer 2

Since one has the term $\sqrt{b^2-4ac}$ in solution of the quadratic formula, if $b^2-4ac>0$, then the equation has two real solutions (since $\sqrt{b^2-4ac}$ is real, and $b$ and $2a$ are also real). When $b^2-4ac<0$, then the quadratic equation has two complex solutions (since $\sqrt{b^2-4ac}$ is complex imaginary). If $b^2-4ac=0$, then $\sqrt{b^2-4ac}=0$, implying that the solution is $x=\dfrac{-b}{2a}$ (can you see why?).

Answer 3

If $ b^2 - 4ac > 0 $ then you will get 2 real roots of the quadratic equation.

If $ b^2 - 4ac < 0 $ then your under root part will become negative and so the equation will have 2 complex roots.

If $ b^2 - 4ac = 0 $ then you will have one solution which will be $ -b/a $

this might help you understand better