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Completing the Square


Date: 09/08/97 at 17:38:59
From: Kelly Shea
Subject: High school algebra (completing the square)

2x^2 = x + 5


Date: 09/19/97 at 13:30:11
From: Doctor Chita
Subject: Re: High school algebra (completing the square)

Hi Kelly:

You sure don't waste any words, do you? I assume you want to know 
how to complete the square to solve this problem, right? Let's work 
through an example: 3x^2 = 6x + 12 

First, to solve a quadratic equation in one variable like this one 
by completing the square, get just the quadratic term and the linear 
term (if there is one) on one side of the equation. Here, subtract 
6x from both sides to get:

      3x^2 - 6x = 12

Second, divide each term in the equation by the coefficient of the 
squared term so its new coefficient is 1.

       x^2 - 2x = 4

Third, divide the coefficient of the linear term (the x-term) by 2 
and square the result. 

       (-2/2)^2 = 1 

Then add this number to both sides. Now you have:

   x^2 - 2x + 1 = 4 + 1

Fourth, simplify the right side:

    x^2 -2x + 1 = 5

The left side is now a perfect square trinomial. Therefore, you 
can factor it into the binomial squared, (x+1)^2 , and write the 
equation as follows.

        (x-1)^2 = 5

Fifth, take the square root of both sides of the equation, noting 
that the square root of a number can be positive or negative.

          (x-1) =  sqrt(5)

Sixth (and lastly) separate and solve each linear equation.

          (x-1) = sqrt(5)      or  (x-1) = -sqrt(5)

              x = sqrt(5) + 1  or      x = -(sqrt5) + 1

I'll use the same steps in this problem. See if you can follow each 
step:

        5x^2 -1x - 10 = 0

            5x^2 - 1x = 10

           x^2 - 1/5x = 2

(Here's the tricky part: divide -1/5 by 2 and square the result. 
Add the number to both sides.)

   x^2 - 1/5x + 1/100 = 2 + 1/100

         (x - 1/10)^2 = 201/100

           (x - 1/10) =  sqrt(201/100)
                      =  sqrt(201)/10

             x - 1/10 = sqrt(201)/10         or 
             x - 1/10 = -sqrt(200)/10

                    x = sqrt(201)/10         or 
                    x = (-sqrt(201)/10)

                    x = sqrt(201)/10 + 1/10  or
                    x = -sqrt(201)/10 + 1/10

                    x = (sqrt(201) + 1)/10   or
                    x = (-sqrt(201) + 1)/10

This solution is not exactly pretty, but it's correct.

The point of this method is to create a perfect square trinomial 
on one side of the equation. Then you can factor it into the form 
(x + k)^2 and take the square roots of both sides of the equation, 
stripping away the square term. What happens on the right with the 
numbers, just happens. When you're done, you may end up with two 
different real values for x, two equal real values for x, or two 
different complex numbers, a subject for another day. 

-Doctor Chita,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Basic Algebra

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