Factoring Quadratics

Date: 06/10/97 at 06:04:33
From: Sebastian Nieto
Subject: Factoring Trinomials into Binomials

I'm having trouble factoring quadratics, such as 2x^2 - 11x + 15 or 
6a^2 + 7a - 5. I probably would've understood this by now, but 
unfortunately the 'explanations' in my math book are terrible.

Date: 06/10/97 at 19:27:31
From: Doctor Anthony
Subject: Re: Factoring Trinomials into Binomials

The quick method is really trial and error, but with intelligent 
guesses.

We'll start by looking at 2x^2 - 11x + 15.

We know the brackets must look like this: (2x   )(x     )

Then we know the signs must be alike (because of the +15) and the 
signs must both be negative because of the -11.

We can write the brackets as: (2x -  )(x -  )

The last terms in the brackets must be factors of 15, say 3 and 5, but 
we must decide where the 3 goes and where the 5 goes.

Now look at the middle term -11x. This is obtained by combining the 
two outer terms in the brackets with the two inner terms.

It is immediately obvious that 2 x 3 + 5 x 1  =  11

We complete the factors as: (2x - 5)(x - 3)

To check, we combine the two outer terms 2 x 3 = 6
                with the two inner terms 5 x 1 = 5
                                              -----
                                         Total  11  

Now let's look at 6a^2 + 7a - 5.        

Step (1)    (3a     )(2a      )
Step (2)    (3a    5)(2a     1)   or the other way round
Step (3)    (3a +  5)(2a  -  1)   or the other way round

This happens to be correct because combining two outers 3 x -1 =  -3
                                        with two inners 5 x  2 =  10
                                                                 -----
                                                          Total =  7

So the correct factors are: (3a + 5)(2a - 1).  
           
With practice you will find that you can write down the factors by 
inspection.

-Doctor Anthony,  The Math Forum
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