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Q&A #104

Teachers' Lounge Discussion: Factoring trinomials

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From: Daniel Springer <daniel_springer@fresnounified.org>
To: Teacher2Teacher Public Discussion
Date: 2009020319:44:24
Subject: trinomial factoring redered trivial by simple method (Can even be done mentally)

I have never seen this in Mathematics Teacher or any other source.

The method renders mentally solving (or factoring) reasonable
quadratics nearly trival.
 
EXAMPLE:
 
If,
           20x^2  +  28x -  3 =  0
 
THINK:  
                  "product = 60, differ by 28, want smaller number"
AHA!
                   "2 and 30"
Thus,
               x = 2/20 or -30/20

DONE! 

In general, if
            ax^2  +  bx +  c =  0,  a is positive

Then, 
             ax(ax + b) = -c

We think of a pair of numbers that differ by b whose product is -c,
lets say the pair is {P, Q}, then {-P, -Q} also satisfy the equation.

We decide if ax is larger or smaller than (ax + b).  This is easy, if
b > 0 then ax is smaller, otherwise 
ax is larger.

Suppose that ax is smaller and P is the smaller number, then 

Then, our answer is:

         x = P/a or   x = -Q/a

                

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