Have you ever owned among those Magic 8 Balls? they look choose comically oversized pool balls, yet have a flat window built right into them, so the you deserve to see what"s insidea 20-sided die floating in venomous opaque blue goo. Supposedly, the billiard ball has actually prognostic powers; every you have to do is ask that a question, give it a shake, and slowly, mystically, favor a petroleum-covered seal emerging from an oil spill, the die will increase to the small window and also reveal the price to her question.

You are watching: What does the discriminant tell us

The quadratic equation consists of a Magic 8 round of sorts. The expression b2 - 4ac from in ~ the radical sign is referred to as the discriminant, and it have the right to actually identify for friend how countless solutions a offered quadratic equation has, if friend don"t feel prefer actually calculating them. Since an unfactorable quadratic equation needs a lot of work to deal with (tons that arithmetic abounds in the quadratic formula, and also a entirety bunch of procedures are forced in the completing the square method), it"s often useful to gaze into the mystic past to make certain the equation even has any kind of real number solutions prior to you spend any kind of time in reality trying to uncover them.

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

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

If you acquire a optimistic number, the quadratic will have actually two unique solutions.If you get 0, the quadratic will certainly have specifically one solution, a double root.

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

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

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

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

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

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

Set a = 3, b = -2, and 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, only two imaginary ones.

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Excerpted from The finish Idiot"s guide to Algebra 2004 through W. Michael Kelley. All legal rights reserved consisting of the right of reproduction in totality or in component in any kind of form. Used by arrangement with Alpha Books, a member the Penguin group (USA) Inc.