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Solving Quadratic Equations by Factoring, a la Shmoop.

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Ever since your little brother saw the movie It, he's been terrified of clowns.

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So for his eighth birthday party, you decide to get a clownÉ just to mess with him.

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Unfortunately, the guy who shows up is not the creepy clown you were hoping for.

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Instead, he's got the biggest smile you've ever seen.

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However, he also seems really into algebra. Which isÉ kinda creepy.

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You decide to pursue this angle and see where it goes.

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The clown tells you his smile can be modeled by the equation y equals x squared plus 6x

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minus 16.

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How wide is the clownÕs smile? This looks like a quadratic equation.

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Let's take a look at the equation on a graph. To find the width of the clown's smile in

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inches, we can calculate the distance between the x intercepts or roots of the parabola.

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The x intercepts are where the parabola crosses the x axis, which means y equals 0. So let's

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set y to 0 in our equation. y equals x squared plus 6x minus 16, which

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equals zero.

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To find the x values where y equals 0, we can factor the right side into the form "the

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quantity x plus p times the quantity x plus q."

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We can use FOIL to multiply this out. FOIL stands for First, Outer, Inner, then Last.

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So X times X is X-squared, plusÉ

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X times Q is "Q-X", plusÉ

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P times x is "P-X", plusÉ

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P times Q is "PQ".

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Since "PX" and "QX" are like terms, we can add them together to make the quantity P plus

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Q times X.

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Let's look at our original equation to compare.

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We can see that P plus Q equals 6 and P times Q equals negative 16.

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So first, let's find two numbers that multiply together to give negative 16.

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HereÕs a chart of all the factors of negative 16É

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1, negative 16É negative 1, 16É

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2, negative 8É negative 2, 8É

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4 and negative 4.

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We're looking for a "P plus Q" value of 6, which only works for 8 and negative 2.

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That means X squared plus 6X minus 16 can be factored to:

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x + 8É timesÉ x Ð 2 For the equation to equal zero, either X plus

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8 or X minus 2 must equal zero.

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Which means x = -8 and x = 2 So our parabola goes through the x-axis at

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points "negative 8, zero" and "2, zero".

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How does this relate to the clown's smile?

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Well, the width of his smile is the distance between those two points, which is 2 minus

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negative 8É or 10 inches. OkayÉ you decide this clown is definitely

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creepy enough for your little brother.

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However, your plan backfires. Instead of turning him off clowns even moreÉ

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Éyour prank turns him ONTO algebra.

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Great. And he was already the good-looking one.

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