Completing the Square: Alternate Method

Date: 02/06/98 at 12:30:16
From: Han-Han Wang
Subject: New method for solving by completing the square 

I have come up with a new method for solving quadratic equations by 
ccompleting the square. Instead of dividing by the coefficient of the 
quadratic term (and getting complicated fractions), I multiply the 
equation to get a perfect square coefficient for the quadratic term. 

I want some feedback. I am sure there are some exceptions to this 
method. Otherwise schools would teach this, right? 

Thanks, 
Hanhan Wang

Date: 02/06/98 at 16:22:00
From: Doctor Rob
Subject: Re: New method for solving by completing the square

This is a valid, alternate method, which is taught in some schools.  
It works as-is if the linear term has an even coefficient. To make 
this method fraction-free, if the coefficient of the linear term is 
odd, you should multiply not by the coefficient of the quadratic term, 
but by four times it. That will insure that the linear term has an 
even coefficient, which is necessary to avoid fractions further. In 
fact, if you always use four times the coefficient of the quadratic 
term, you will always avoid fractions. Example:

    3*x^2 - 7*x - 20 = 0,
   9*x^2 - 21*x - 60 = 0,
       (3*x - 7/2)^2 = 60 + 49/4 = 289/4 = (17/2)^2

See that fractions are still present. If instead of multiplying by 3, 
we multiply by 4*3, we will still get a perfect square coefficient of 
the quadratic term:

 36*x*2 - 84*x - 240 = 0,
         (6*x - 7)^2 = 240 + 49 = 289 = 17^2,

and you can finish the solution

   6*x - 7 = 17 or -17,
       6*x = 24 or -10,
         x = 4 or -5/3.

Symbolically,

              a*x^2 + b*x + c = 0,
  4*a^2*x^2 + 4*a*b*x + 4*a*c = 0,
                (2*a*x + b)^2 = b^2 - 4*a*c

and so on. You see, if b is even, then a 4 can be divided out of the
last equation, but if it is odd, you can't do that without introducing
fractions.

The disadvantage of using this method is that the integers involved 
can get fairly large. You may be faced with taking the square root of 
a four- or five-digit number. The advantage, as you point out, is that 
you avoid working with fractions.

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