ShmoopTube

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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. [young brother watching IT on TV]

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

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Unfortunately, the guy who shows up is not the creepy clown you were hoping for. [playful clown arrives at the front door]

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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.

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Which is… 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 [clown describing his smile with an equation]

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minus 16.

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How wide is the clown’s smile?

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This looks like a quadratic equation.

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Let's take a look at the equation on a graph. [quadratic equation shown on a graph]

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To find the width of the clown's smile in inches, we can calculate the distance between

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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.

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So let's set y to 0 in our equation. [y set as 0 in the quadratic equation]

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0 equals x squared plus 6x minus 16,

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To find the x values where y equals 0, we can factor the right side into the form "the [right side of the formula highlighted]

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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. [aluminum foil used to multiply the equation]

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FOIL stands for First, Outer, Inner, then Last.

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So X times X is X-squared, plus…

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X times Q is "Q-X", plus…

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P times x is "P-X", plus…

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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 [PX and QX highlighted as like terms]

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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. [clown combines two question marks together and -16 appears above his head]

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Here’s a chart of all the factors of negative 16…

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1, negative 16… negative 1, 16…

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2, negative 8… negative 2, 8…

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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 + 8… times…

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x – 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 [Parabola shown going through the x axis]

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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 [Clown smiles and person uses tape measure to measure the width]

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negative 8… or 10 inches.

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Okay… you decide this clown is definitely creepy enough for your little brother. [older brother allows clown to enter]

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However, your plan backfires.

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Instead of turning him off clowns even more…

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…your prank turns him ONTO algebra. [clown holding an algebra book for the young brother]

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Great job!

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He was already the good-looking one.

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