Two Methods of Factoring Quadratics

Date: 04/14/98 at 21:44:35
From: Derek
Subject: Factoring quadratic trinomials

I was gone for a week and missed the lesson on how to factor quadratic 
trinomials. Could you please show me an example?

Date: 04/15/98 at 11:35:11
From: Doctor Rob
Subject: Re: Factoring quadratic trinomials

A quadratic trinomial has the form a*x^2 + b*x + c, for some 
expressions a, b, c, and x. In the simplest case, a, b, and c are 
constants, and x is an unknown. For example, you might want to factor 
3*x^2 - 5*x - 2. Here a = 3, b = -5, and c = -2. You want to factor it 
into two linear factors:

     a*x^2 + b*x + c = (r*x+s)*(t*x+u)

where r, s, t, and u are expressions of the same kind as a, b, and c.
In the example, it is true that 3*x^2 - 5*x - 2 = (3*x + 1)*(x - 2).
For the general equation to be true, if you expand the righthand 
side, you get:

     a*x^2 + b*x + c = (r*t)*x^2 + (r*u+s*t)*x + (s*u)

This means that you want the following three equations to be true:

     a = r*t
     b = r*u + s*t
     c = s*u
 
You know a, b, and c, and you want to find r, s, t, and u, if 
possible. In the example, r = 3, s = 1, t = 1, and u = -2, and 
a = 3 = r*t, b = -5 = 3*(-2) + 1*1, and c = -2 = 1*(-2). One way to 
proceed is as follows.

Multiply together a and c. Now try to factor that product into two 
factors whose sum is b. In the example, a*c = 3*(-2) = -6.  You want 
to write -6 as a product of two factor whose sum is b = -5.

Why would this help?  Because a*c = (r*t)*(s*u) = (r*u)*(s*t), and 
b = r*u + s*t. The two factors of a*c whose sum is b are r*u and s*t.
Once you have done this, you can write b as the sum of those two 
factors, and regroup the terms, factor each group, then factor the 
whole:

     a*x^2 + b*x + c = r*t*x^2 + (r*u + s*t)*x + s*u
                     = (r*t*x^2 + r*u*x) + (s*t*x + s*u)
                     = r*x*(t*x + u) + s*(t*x + u)
                     = (r*x + s)*(t*x + u)

In the example, the two factors of -6 which add up to -5 are -6 and 1.
Then:

     3*x^2 - 5*x - 2 = 3*x^2 - 6*x + 1*x - 2
                     = 3*x*(x - 2) + 1*(x - 2)
                     = (3*x + 1)*(x - 2)

Here is another example: factor 2*x^2 - 7*x + 6.

Multiply together 2*6 = 12. Now try to factor 12 into two numbers 
whose sum is -7. The two numbers are -4 and -3. Then:

     2*x^2 - 7*x + 6 = 2*x^2 - 4*x - 3*x + 6
                     = 2*x*(x - 2) - 3*(x - 2)
                     = (2*x - 3)*(x - 2)

If you cannot find a way to factor a*c into two factors that add up 
to b, then the trinomial cannot be factored.

Example:  x^2 + x + 1.  Note that a*c = 1, and you need to find two 
factors of 1 that add up to b = 1. The only pairs of numbers whose 
product is 1 are 1 and 1, or -1 and -1, and the sums are 2 and -2. 
This is not possible, so this expression cannot be factored.

Sometimes the expression I have labeled x is not just a variable. For
example, it may be a power of a variable. To factor t^6 - t^3 - 30, 
we can take x = t^3. Then we have x^2 - x - 30. We need to find a 
factorization of -30 into two numbers whose sum is -1. They are 
-6 and 5, and the result is (x - 6)*(x + 5) = (t^3 - 6)*(t^3 + 5).

Another method is to "complete the square." The idea of this method is 
to express:

     a*x^2 + b*x + c = a*[(x + d)^2 - e]

If e is a perfect square, then we can factor the part inside brackets 
as the difference of two squares. By expanding the righthand side,

     a*x^2 + b*x + c = a*x^2 + 2*a*d* x + a*d^2 - a*e

and to make the "x" terms the same, we need to pick d = b/(2*a). The
procedure goes like this. Figure out d = b/(2*a). Factor a out of all 
terms. Inside the remaining factor, add d^2 to make a perfect square 
(x+d)^2, and subtract d^2 to make the equation still true.  Figure out 
e, and decide if it is a perfect square. If not, the trinomial is not 
factorable. If so, factor the difference of squares. Simplify.

Example:  Factor 3*x^2 - 5*x - 2.  d = -5/6

     3*x^2 - 5*x - 2 = 3*[x^2 - 5/3*x - 2/3]
                     = 3*[x^2 - 5/3*x + 25/36 - 25/36 - 2/3]
                     = 3*[(x - 5/6)^2 - 49/36]

Since e = 49/36 = (7/6)^2 is a perfect square, this is going to work:

                     = 3*[(x - 5/6)^2 - (7/6)^2]
                     = 3*[(x - 5/6) - 7/6]*[(x - 5/6) + 7/6]
                     = 3*(x - 2)*(x + 1/3)
                     = (x - 2)*(3*x + 1)

Example:  Factor x^2 + x + 1.  d = 1/2.

     x^2 + x + 1 = x^2 + x + 1/4 - 1/4 + 1
                 = (x + 1/2)^2 + 3/4

Since e = -3/4 is not a perfect square, this trinomial is not 
factorable.

-Doctor Rob,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/