Have you ever owned one of those Magic 8 Balls? they look favor comically oversized swimming pool balls, but have a flat home window built right into them, so that you can see what"s insidea 20-sided dice floating in disgusting opaque blue goo. Supposedly, the billiard ball has actually prognostic powers; every you need to do is ask the a question, provide it a shake, and also slowly, mystically, prefer a petroleum-covered seal emerging from an oil spill, the dice will increase to the tiny window and also reveal the answer to your question.

You are watching: How many real solutions does a quadratic equation have

The quadratic equation contains a Magic 8 round of sorts. The expression b2 - 4ac from in ~ the radical sign is dubbed the discriminant, and also it deserve to actually identify for girlfriend how countless solutions a given quadratic equation has, if girlfriend don"t feel like actually calculating them. Considering that an unfactorable quadratic equation requires a many work to solve (tons that arithmetic abounds in the quadratic formula, and a entirety bunch of measures are forced in the perfect the square method), it"s often useful to gaze into the mystic past to make certain the equation also has any type of real number solutions prior to you spend any type of time actually trying to discover them.


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The discriminant is the expression b2 - 4ac, i m sorry is identified for any type of quadratic equation ax2 + bx + c = 0. Based ~ above the sign of the expression, you deserve to determine how numerous real number remedies 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 view what results:

If you acquire a positive number, the quadratic will have actually two unique solutions.If you get 0, the quadratic will have exactly one solution, a twin root.

The discriminant isn"t magic. It simply shows how necessary that radical is in the quadratic formula. If that radicand is 0, because that example, climate you"ll get

a single solution. If, however, b2 - 4ac is negative, climate you"ll have actually a an unfavorable inside a square root sign in the quadratic formula, meaning only imagine solutions.

Example 4: without calculating them, recognize how many real services the equation 3x2 - 2x = -1 has.

Solution: collection the quadratic equation same to 0 by including 1 come both sides.

3x2 - 2x + 1= 0
You"ve obtained Problems

Problem 4: there is no calculating them, recognize how numerous real remedies the equation 25x2 - 40x + 16 = 0 has.


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

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

Because the discriminant is negative, the quadratic equation has no genuine number solutions, just two imagine ones.

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Excerpted from The finish Idiot"s guide to Algebra 2004 by W. Michael Kelley. All civil liberties reserved including the right of reproduction in totality or in part in any form. Used by setup with Alpha Books, a member of Penguin group (USA) Inc.