Solving Quadratic Equations by Factoring
How else would you solve a quadratic equation? Oh, that's right: a zillion other ways.
|Quadratic Equations||Quadratic Equations|
However, he also seems really into algebra.
Which is… kinda creepy.
You decide to pursue this angle and see where it goes.
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]
How wide is the clown’s smile?
This looks like a quadratic equation.
Let's take a look at the equation on a graph. [quadratic equation shown on a graph]
To find the width of the clown's smile in inches, we can calculate the distance between
the x intercepts or roots of the parabola.
The x intercepts are where the parabola crosses the x axis, which means y equals 0.
So let's set y to 0 in our equation. [y set as 0 in the quadratic equation]
0 equals x squared plus 6x minus 16,
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]
quantity x plus p times the quantity x plus q."
We can use FOIL to multiply this out. [aluminum foil used to multiply the equation]
FOIL stands for First, Outer, Inner, then Last.
So X times X is X-squared, plus…
X times Q is "Q-X", plus…
P times x is "P-X", plus…
P times Q is "PQ".
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]
Q times X.
Let's look at our original equation to compare.
We can see that P plus Q equals 6 and P times Q equals negative 16.
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]
Here’s a chart of all the factors of negative 16…
1, negative 16… negative 1, 16…
2, negative 8… negative 2, 8…
4 and negative 4.
We're looking for a "P plus Q" value of 6, which only works for 8 and negative 2.
That means X squared plus 6X minus 16 can be factored to:
x + 8… times…
x – 2 For the equation to equal zero, either X plus
8 or X minus 2 must equal zero.
Which means x = -8 and x = 2 So our parabola goes through the x-axis at [Parabola shown going through the x axis]
points "negative 8, zero" and "2, zero".
How does this relate to the clown's smile?
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]
negative 8… or 10 inches.
Okay… you decide this clown is definitely creepy enough for your little brother. [older brother allows clown to enter]
However, your plan backfires.
Instead of turning him off clowns even more…
…your prank turns him ONTO algebra. [clown holding an algebra book for the young brother]
He was already the good-looking one.