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3:3 Factoring Quadratic Trinomials

Page history last edited by Math in a Box - Susan Johnsey gm 3 years, 3 months ago
 

LESSON 3.3    Factoring Quadratic Trinomials   ax2+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 x2, another 

x, and the third term has no variable at all. There is a number in each of the three terms. 

 

ax2+ 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):

 

    3x2+ 6x -4,  thus

a = 3

b = 6

c = -4

  

x2- 8x +15, thus

 

a = 1 

b =-8  

c =15

BEWARE:

 -4-2x2-6x , thus

 

a = -2 be careful

b = -6

c= -4

 

x2- 8x , thus

 

a = 1 

b =-8

c = 0

 

9 - x2 , 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:  x2 - 6x+8 equals what 2 binomials multiplied together?

 

The clue is to look at the x2 as x times x and then to look at the +8.

 

Yes , JUMP over the -6x for now.

     x2- 6x + 8 = (x - 4)(x - 2)  These are the two binomials. 

You may also write it as   x2- 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 x2- 6x + 8. 

 

EXAMPLE 2:  MY PROBLEM: x2-  9x + 8 

Look at the x2 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.

x2 - 9x + 8 =  See answer below by &&&&& at bottom of this lesson.

EXAMPLE 3: MY PROBLEM: x2 +12x + 36 

x2 + 12x +36 = See answer below by &&&&&

 

EXAMPLE 4: MY PROBLEM: x2 - 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?     x2- 6x - 27  is same as 1x2- 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:

http://www.youtube.com/embed/qh8sYcKyrS0

 

 

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   x2 +17x+60 and

                          y 2+17y+60 or

                          m2 +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:

 

x2 +17xy+60y2

 

=(x  + 5y)(x + 12y)   the last two terms give the 60y2but also they will help make the 17xy term

x2 +12xy+5xy+60y2    =

  x2 +17xy+60y2

EXAMPLE 6:

 

       k2 +7kp+10p2

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

k2 +5kp+2kp+10p2 =

 

  k2 +7kp+10p2

 

EXAMPLE 7:

 

      p2 - 12pxh+35x2h2

 

=(p -7xh)(p -5xh)

p2 - 5pxh -7pxh+35x2 h2 =

 

p2 - 12pxh+35x2 h2

Now try these:

 

EXAMPLE 8:  x2-11xt+30t2

 

EXAMPLE 9:  r2+12ryx - 28y2x2  

 

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 x2 in the quadratic trinomials) always = 1.  

 

USE the BOX in the video to complete these problems.   

 

     2x2 +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  2x2+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:  3x2 -10x - 8   Watch the negatives.

 

  EXAMPLE 12: Also FACTOR   5x2 -7xy -6y2 and 

 

  EXAMPLE 13  :    FACTOR  12x2n + xn -1  

  Do you recall that xn times xn is x2n ?  

          

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 @@@.

7x2 - 11x -6   and        

12x 2- 56x + 9 and

16x2- 25     (hint the x term is +0x)

 

 

&&&&&  EASY ONES:

 EX.2     1x2 -9x+8 =

(x-1)(x-8) OR (x-8)(x-1)

 

 EX.3    x2+12x +36=

(x + 6)(x + 6)

 

 EX.4    x2 - 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 (4xn - 1)(3xn + 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)(6x-1) 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.

 

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