Binomials: Completing the Square

Date: 08/01/97 at 20:07:23
From: Anonymous
Subject: Binomials

Dr. Math,

I have two questions I really need your help on!

1. Find two binomials whose product is X ^ 2 - 25.

2. Find two binomials whose product is X ^ 2 - 6X + 9.

Could you please tell me how you decided on these two binomials for 
each question?

Thank you in advance for your help,
BradyRed

Date: 08/07/97 at 15:27:22
From: Doctor Rob
Subject: Re: Binomials

Problems like this always depend on the single fact that 
A^2 - B^2 = (A + B)*(A - B) for any expressions A and B.

The first problem is already in that form, with A = X and B = 5.

The second problem needs to be manipulated a little to put it into 
that form.  First factor out the coefficient of X^2 from every term 
in the given expression (in this case it is 1, so nothing needs to be 
done).

Then take the remaining part, and try to write it in the form
(X + b)^2 - d^2 = X^2 + 2*b*x + b^2 - d^2.  The coefficients of X 
must be the same, and the constant terms must be the same, so 
2*b = -6 and b^2 - d^2 = 9. Solving this tells us that b = -3 
and d = 0.  Then

  X*2 - 6*X + 9 = (X - 3)^2 - 0^2.

Now we use the fact first given above with A = X + b and B = d, and
out pop the two binomials, X + b + d and X + b - d.

The procedure outlined in the last paragraph above is called 
"completing the square".

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