Solving Equations by Factoring
I have created a free course for learning to factor. There are at least 5 major types to learn.
You will see factoring in many courses after Algebra 1 so be sure you can factor polynomials.
A link to the course is on the WIKI tab with the listing of my other courses. CHECK it out.
Do you remember multiplying with ZERO: 5(0)=0 or 0(9)=0 or 0(0)=0.
That's easy!!
It is simply arithmetic, but it is important in our next use of factoring. I hope you recall factoring.
We have had several lessons on that topic.
Factoring is writing an algebra expression as MULTIPLICATION.
When we multiply with zero our answer is always zero.
Do you recall 0 times 0=0 or that 5 times 0 = 0 ?
No other number has that quality.
If (x5) times (x+9)=0 then x5=0 or x+9=0 or both equal 0!
Right?
Thus (x5)(x+9)=ZERO means that x 5=0 or that the x+9=0.

NOW Solve the above two algebra equations:
x5=0
so x= +5

x+9=0
so x= 9.

x=5 and x=9 are the solutions for (x5)(x+9)=0 .
Check them by letting x=5:
(x5)(x+9)=
(55)(5+9)=
(0)(14)=0 .
TRUE!
Now you check the 9.
It too will make our equation true.
Solve (x5)(x+9)=0
means x= 5 or x=9 .
These are called solutions.

THESE are the EASY ONES!
CONSIDER x^{2}+4x  45 =0.
Do you know where I got this quadratic trinomial?
I hope so. If not then multiply (x5)(x+9). YOU must know how; some call it FOIL others call it the distributive property.
(x5) times (x+9) = x^{2}+9x5x45 simplifies to x^{2}+4x 45.
If I take the trinomial x^{2}+4x45 and substitute 9 for each x then
I will have (9)^{2}+4(9)45
=813645 which is 0.
If I take the trinomial x^{2}+4x  45 and substitute a 5 for each x then
I will have (5)^{2}+4(5)45
=25+2045 which is 0.

So you see if we have (x5)(x+9) or x^{2}+4x45 equals ZERO
then x=9 or x=5.
These are the only numbers that will make (x5)(x+9) or x^{2}+4x45 equal ZERO.

Solve x^{2}+4x45=0. The values of x that make this true were 5 and 9.
We call those numbers the solution to x^{2}+4x45.
Let us solve some polynomial equations. Here are the steps that we used above. We will : 
1. rewrite equation = 0 (must equal 0),

2. factor the expression (written as multiplies),

3. solve each of the simple equations for x.
( WE will have several linear equations to solve.)

EXAMPLE: x^{2} 17x +72 = 0
becomes (x 9)(x 8) =0.
WE Factored.
we have x9 =0 and x8 =0. Now solve for x.
Solutions are x=9 and x=8.
You try these. Write these in your notebook so that you have good examples to follow.
t^{2}  12 = 4t
We must begin by rewriting.

18m^{2}  8 = 0 Note the common factor of 2.
2(9m^{2}  4) =0

t^{2}  4t 12 = 0

2(3m 2)(3m+2) = 0

(t 6)(t +2) = 0

3m 2 =0 or 3m+ 2 =0

t  6 =0 or t+2 = 0

3m = 2 or 3m = 2

t= 6 or t= 2
two solutions

m = 2/3 or m =  2/3
two solutions

I have created two videos for factoring.
Please visit my web site www.mathinabox.com and click the tab "Lessons / Videos".
You will see the two videos listed for FACTORING QUADRATICS .
Click here to go to http://www.mathinabox.com/LessonsVideos.html