Rule for Completing the Square

Date: 03/15/2002 at 12:08:28
From: Irina
Subject: Square equation

Which of the following expressions should be placed in each set of 
parentheses below in order to solve the equation by completing the 
square?

x^2+6x+(?)=15+(?)

A.3/2
B.3
C.6
D.9

I have no clue to answer this question.
Thanks.

Date: 03/15/2002 at 14:52:03
From: Doctor Rick
Subject: Re: Square equation

Hi, Irina.

The idea is that you have the equation

  x^2 + 6x = 15

and you want to solve it by the method of completing the square. This 
means that you want to make the left side into the square of a 
binomial, which you can do by adding the correct constant to it; and 
of course, to keep the equation equivalent to the original, you must 
add the same constant to the right side as well.

How do you tell what constant to add? We want to end up with something 
like

  (x + C)^2

Expand this:

  x^2 + 2Cx + C^2

Compare with what you have:

  x^2 + 6x + ___

The coefficient of x, 2C, should be equal to 6:

  2C = 6
  C = 6/2 = 3

Then what goes in the blank? It's C^2, and since C = 3, it's 3^2 = 9 
that goes in the blank.

  x^2 + 6x + (9) = 15 + (9)

Then we can solve the problem:

  (x + 3)^2 = 24
  x + 3 = sqrt(24) or -sqrt(24)
  x = -3+sqrt(24) or -3-sqrt(24)

You can simplify the square root of 24, but that's another topic.

Now that we've worked one problem out in detail, we can see a simple 
rule for completing the square - at least if the coefficient of x^2 
is 1, as in this example. Just take half the coefficient of x, and 
square it.

  (6/2)^2 = 9

If you have any problems in which the coefficient of x is not 1, then 
work it out in the same way I did this example, and see if you can 
find the rule for this more general case.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/   

Date: 03/18/2002 at 11:07:11
From: Irina
Subject: Square equation

Thanks a lot; that information was very helpful.
Irina