Algebra (Period 1)

Solving Equations By Factoring 6-8

Zero Products Property -- a "friend" you can count on..."
For any two rational numbers a and b, if ab = 0, then either a = 0 or b = 0 or both equal zero

How does this help you?

Try solving
x² + 10x + 24 = 0
Your first reaction is to subtract 24 from both sides you would get
x² + 10x = -24 BUT NOW WHAT?????
It's a quadratic ( x² term). You can't isolate x because there is another term with the x² in it. THe Zero Products Property will help you solve quadratics.

Using this property we can solve quadratic equations by factoring the equation and setting each factor to zero and solving them.

x² + 10x + 24 = 0
factor as usual-- using your rules from the other sections:
(x +6) (x +4) = 0
set each one to zero
x + 6 = 0 and x + 4 = 0
so x = -6 and x = -4

Plug them back-- and it works!!

IF you are already given it factored--just set each to zero
(5x +1)(x-7) = 0
Using the Zero Products Property we know either 5x - 1 must be equal to zero or x - 7 must equal zero... that's the only way that the product could be zero...
Set both equal to zero and solve.
if 5x + 1 = 0 then x = -1/5 ( its just a 2 step equation)
if x - 7 = 0 then x = 7

If you substitute these answers for x back in the original equation, they will both end up as 0 = 0 which is what you want. The left side equals the right side!!

In the second example, the equation was already factored.

WIth the first example, you saw you needed to first factor the equation ( if possible) and then set each factor to zero to solve.
MAKE SURE YOU MOVE EVERYTHING TO ONE SIDE OF THE EQUATION. YOU MUST HAVE ZERO ON ONE SIDE OF THE EQUATION TO USE THE ZERO PRODUCTS PROPERTY

x² = 16

First move the 16 to the left side
x² -16 = 0
Now... you have the difference of two squares
(x + 4)(x-4) = 0
Set each factor to zero and solve
x + 4 = 0 and x - 4 = 0
so x = -4 and x = 4

Why do you have to have zero on one side ?

NEW ALGEBRA TERM
root-- any solution that turns the equation into the value of zero is called a root of the polynomial, or a ZERO of the polynomial because when you are graphing the polynomial, the y value is zero at this point.
So if the directions say " FInd the roots of..." it just means get zero on one side of th quation, factor, set each factor to zero and solve.

Summary of solving quadratics
solutions = roots= eros of the polynomials

1. Get zero on one side of the equation
2. Factor
3. Set each piece ( factor) equal to zero
4. Solve as one or two step equations
5. Check by substituting in the ORIGINAL equation