ShmoopTube

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Using the Discriminant, a la Shmoop. It's never okay to discriminate…

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…but you can definitely use the discriminant to help answer some of life's most difficult

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questions. In the quadratic formula b2 – 4ac is the

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discriminant.

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The discriminant determines the nature and number of solutions of a quadratic equation.

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If b2 – 4ac is positive and greater than zero, there will be two distinct, real solutions.

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If b2 – 4ac is zero, there will be one distinct, real solution.

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But, if b2 – 4ac is negative and less than zero, there are no real solutions.

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None. Don't even try to make one up. It won't happen.

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Let's get quadratic for a couple of examples.

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Our first example is x-squared plus three-x plus five equals zero.

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Plugging these bad boys into the discriminant, b squared minus 4ac, we get three-squared

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minus four-times-one-times-five.

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This simplifies to nine minus twenty, which is negative eleven.

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This is less than zero…there are no real solutions.

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What does “no real solutions” mean? If we take a look at the graph, the parabola

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does not touch the x-axis at any x-value.

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For our second example, we have the equation...

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x-squared minus six-x minus ten equals zero.

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Plugging this into the discriminant it looks like...

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six-squared minus four time negative-ten times one.

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This is thirty-six plus forty, which is seventy-six.

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Seventy-six is greater than zero. So there are two distinct, real solutions to this problem.

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By looking at the graph, we can see that there are two real solutions because the parabola

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touches the x-axis at two distinct x-values. Okay, so the Magic Discriminant Ball might

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not be able to help you with some of life's biggest questions…

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… but using the discriminant can help you ace your next math test.

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