LESSON 3.3 Factoring Quadratic Trinomials ax^{2}+bx+c
To return to the Home page just click the "Wiki" tab above.
Factoring Quadratic trinomials is very important . What are they?
One Example of a very simple Quadratic: They have 3 terms one of which has x^{2}, another
x, and the third term has no variable at all. There is a number in each of the three terms.
ax^{2}+ bx + c where a ,b ,and c represent numbers is the general form of a quadratic trinomial.
The a and b and c are the numbers. The "a" can not be zero, but the b and c could be.
Here are some examples of quadratic trinomials (one variable):
3x^{2}+ 6x 4, thus

a = 3
b = 6
c = 4

x^{2} 8x +15, thus

a = 1
b =8
c =15

BEWARE:
42x^{2}6x , thus

a = 2 be careful
b = 6
c= 4

x^{2} 8x , thus

a = 1
b =8
c = 0

9  x^{2} , thus

a =  1
b = 0 watch out!
c = 9

There are several methods used to teach factoring quadratics.
I will use the guess method on this first examples, but if
you do not like guessing and want another method then watch the video below
or email me and ask about your method or your school's method.
The most common is trial and error which involves guessing and then checking by using FOIL.
If you've not studied FOIL, don't worry about it.
You know it by the name of "multiply two binomials" which we studied in Section 2.
The first 4 examples below are the easy ones because a=1. So do not stop there!
Learn these and then we will learn what to do when a does not =1.
Or just skip these first 4 examples where a=1 and learn the others first. I prefer that actually.
EXAMPLE 1: x^{2}  6x+8 equals what 2 binomials multiplied together?
The clue is to look at the x^{2} as x times x and then to look at the +8.
Yes , JUMP over the 6x for now.
x^{2} 6x + 8 = (x  4)(x  2) These are the two binomials.
You may also write it as x^{2} 6x + 8 = (x  2)(x  4) .
There is only 1 correct answer when factoring completely. You can write the factors in different order.
So (x  2)(x  4) or (x  4)(x  2) are the answers for the factoring of x^{2} 6x + 8.
EXAMPLE 2: MY PROBLEM: x^{2} 9x + 8
Look at the x^{2} as xx, that is x times x and then look at +8.
And as we did above we skip over the 9x for now.
If you are having trouble with these (it is done by guessing) then please watch my video below.
x^{2}  9x + 8 = See answer below by &&&&& at bottom of this lesson.
EXAMPLE 3: MY PROBLEM: x^{2} +12x + 36
x^{2} + 12x +36 = See answer below by &&&&&
EXAMPLE 4: MY PROBLEM: x^{2}  6x  27. See answer below by &&&&&
Your signs must be exactly as the answers. Watch the signs !
There is a video below that I created.
It will help you with these and the problems for Assignment 3.3B
The above examples are the easy ones so watch the video
if you are not sure you can guess the answers
for ones like: 12x ^{2} 56x + 9 . See the a = 12.
The easy ones above have a =1.
Do you see that? x^{2} 6x  27 is same as 1x^{2} 6x  27.
MANY STUDENTS really like my video method especially for the hard ones.
If you do not see the video above you can view it at Youtube:
Do Assignment 3.3a
Don't continue until you know the above well.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
In the above problems I kept the problems simple in two ways.
First way, I used only the variable x, but any variable can be used.
The problems x^{2} +17x+60 and
y ^{2}+17y+60 or
m^{2} +17m+60 are essentially the same.
But more than one variable can be used in a problem.
Study these carefully .
Do you recall that kp=pk ,
AND pxh = xph = hpx.
HERE are EASY ONES but with more than one variable:
My Problem

My Guess and Result

YOU MUST Multiply to Check

EXAMPLE 5:
x^{2} +17xy+60y^{2}

=(x + 5y)(x + 12y) the last two terms give the 60y^{2}but also they will help make the 17xy term

x^{2} +12xy+5xy+60y^{2} =
x^{2} +17xy+60y^{2}

EXAMPLE 6:
k^{2} +7kp+10p^{2}

=(k +2p)(k +5p)

k^{2} +5kp+2kp+10p^{2} =
k^{2} +7kp+10p^{2}

EXAMPLE 7:
p^{2}  12pxh+35x^{2}h^{2}

=(p 7xh)(p 5xh)

p^{2}  5pxh 7pxh+35x^{2} h^{2} =
p^{2}  12pxh+35x^{2} h^{2}

Now try these:
EXAMPLE 8: x^{2}11xt+30t^{2}
EXAMPLE 9: r^{2}+12ryx  28y^{2}x^{2}
See answers below by $$$$
Now the second way I made the first assignment simple is by keeping
the "a" (remember that is the number in front of the x^{2} in the quadratic trinomials) always = 1.
USE the BOX in the video to complete these problems.
2x^{2} +7x+ 3
The box is a very good way to complete these.
There are several methods that are used in school.
If you know those it is ok to use them,
but if not then please use the BOX in the video above.
EXAMPLE 10: FACTOR 2x^{2}+7x+ 3 Write 2x and x under the F in the box above.
Now write +1 and +3 under the L. Do you know where to write the 3?
do you know if this will give the 7x? Watch the video to find out.
(2x+1)(x+3)
EXAMPLE 11: Now FACTOR this one: 3x^{2} 10x  8 Watch the negatives.
EXAMPLE 12: Also FACTOR 5x^{2} 7xy 6y^{2} and
EXAMPLE 13 : FACTOR 12x^{2n} + x^{n} 1
Do you recall that x^{n} times x^{n} is x^{2n} ?
Here is a game to play. You will need to understand the Examples 11 and 12 well.
GAME Can you get 100% on the first guess?
See answers below by ^^^^^.
BE SURE you can do these well!!!
You will get stuck on this section if you can not.
Also try these and find answers below by @@@.
7x^{2}  11x 6 and
12x ^{2} 56x + 9 and
16x^{2} 25 (hint the x term is +0x . )
&&&&& EASY ONES:
EX.2 1x^{2} 9x+8 =

(x1)(x8) OR (x8)(x1)

EX.3 x^{2}+12x +36=

(x + 6)(x + 6)

EX.4 x^{2}  6x  27 =

(x + 3)(x  9) OR (x  9)(x + 3)



$$$$ (x  5t)(x  6t) and (r +14yx)(r  2yx)
^^^^^.
(x  4)(3x +2) and (x  2y)(5x + 3y) and (4x^{n}  1)(3x^{n} + 1)
These do take practice( finding the correct two numbers for the x term), but everyone does finally get the hang of it.
@@@
(7x + 3)(x 2) and (2x 9)(6x1) and (4x 5)(4x + 5)
This is a very important lesson.
We will continue to use these techniques throughout this class.
If you have trouble, draw a BOX like the one above. Complete it and then do the factoring.
MAKE a special note for yourself (where you can find it later)
about EXAMPLE 13 above.
Do Assignment 3.3b.
Comments (0)
You don't have permission to comment on this page.