How to know how many solutions an equation has

All Signs Point to the Discriminant

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. 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.

The quadratic equation contains a Magic 8 Ball of sorts. The expression b2 - 4ac 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. 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.

Talk the Talk

The discriminant is the expression b2 - 4ac, which is defined for any quadratic equation ax2 + bx + c = 0. Based upon the sign of the expression, you can determine how many real number solutions the quadratic equation has.

Here's how the discriminant works. Given a quadratic equation ax2 + bx + c = 0, plug the coefficients into the expression b2 - 4ac to see what results:

• If you get a positive number, the quadratic will have two unique solutions.
• If you get 0, the quadratic will have exactly one solution, a double root.
• If you get a negative number, the quadratic will have no real solutions, just two imaginary ones. (In other words, solutions will contain the i you learned about in Wrestling with Radicals.)

The discriminant isn't magic. It just shows how important that radical is in the quadratic formula. If its radicand is 0, for example, then you'll get

a single solution. If, however, b2 - 4ac is negative, then you'll have a negative inside a square root sign in the quadratic formula, meaning only imaginary solutions.

Example 4: Without calculating them, determine how many real solutions the equation 3x2 - 2x = -1 has.

Solution: Set the quadratic equation equal to 0 by adding 1 to both sides.

• 3x2 - 2x + 1= 0

You've Got Problems

Problem 4: Without calculating them, determine how many real solutions the equation 25x2 - 40x + 16 = 0 has.

Set a = 3, b = -2, and c = 1, and evaluate the discriminant.

• b2 - 4ac
• =(-2)2 - 4(3)(1)
• = 4 - 12
• = -8

Because the discriminant is negative, the quadratic equation has no real number solutions, only two imaginary ones.

Excerpted from The Complete Idiot's Guide to Algebra © 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.

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