It just shows how important that radical is in the quadratic formula. (In other words, solutions will contain the i you learned about in Wrestling with Radicals.)

If you get a negative number, the quadratic will have no real solutions, just two imaginary ones.

If you get 0, the quadratic will have exactly one solution, a double root.

If you get a positive number, the quadratic will have two unique solutions.

Given a quadratic equation ax 2 + bx + c = 0, plug the coefficients into the expression b 2 - 4 ac to see what results: Based upon the sign of the expression, you can determine how many real number solutions the quadratic equation has.

The discriminant is the expression b 2 - 4 ac, which is defined for any quadratic equation ax 2 + bx + c = 0. Considering that an unfactorable quadratic equation requires a lot of work to solve (tons of arithmetic abounds in the quadratic formula, and a whole bunch of steps are required in the completing the square method), it's often useful to gaze into the mystic beyond to make sure the equation even has any real number solutions before you spend any time actually trying to find them. The expression b 2 - 4 ac from beneath the radical sign is called the discriminant, and it can actually determine for you how many solutions a given quadratic equation has, if you don't feel like actually calculating them.

The quadratic equation contains a Magic 8 Ball of sorts. Supposedly, the billiard ball has prognostic powers all you have to do is ask it a question, give it a shake, and slowly, mystically, like a petroleum-covered seal emerging from an oil spill, the die will rise to the little window and reveal the answer to your question. Have you ever owned one of those Magic 8 Balls? They look like comically oversized pool balls, but have a flat window built into them, so that you can see what's insidea 20-sided die floating in disgusting opaque blue goo. Solving One-Variable Quadratic Inequalities.

Discriminant of a quadratic equation