Sunday, July 21, 2013

QUICK STEPS TO COMPLETE THE SQUARE

This article is going to be a “quickie” because I believe Completing the Square should be just that....quick, easy.  So let’s dispose of all the middle steps and just move swiftly from the General Quadratic Equation to the Vertex Form.

We’ll use a sample equation.......12X^2 + 8X - 10 = Y

of the Vertex Form.................................A(x + h)^2 + K = Y

2.  From the original equation,
think of the X-variable
terms as a separate entity......(12X^2 + 8X) - 10 = Y

coefficient, so that the
coefficient on the X^2
is just 1..............................12(X^2 + 8/12 X) - 10 = Y

I’d reduce that improper
fraction...............................12(X^2 + 2/3 X) - 10 = Y

4.  Fill in “A” in the
Vertex Form..........................................12(X + h)^2 + K = Y

5.  Fill in the “h” value
of the Vertex Form
with 1/2 the new
coefficient on X in
Step 3..............................................12(X + 2/6)^2 + K = Y

I like smaller numbers, so I’m reducing
that pesky improper fraction................12(X + 1/3)^2 + K = Y

6.  Think for a second about what we’ve
done.  By creating the (X + h) binomial
and squaring it, we’ve actually added
more to the equation.  (FOIL it through
if you need proof.)  So we need to
remove it again.

Square “h”, multiply it by “A”,
and subtract it from the
constant in the General
Form equation...................12(X + 1/3)^2 + (- 10 - 12/9) = Y

I feel I’m doing more reducing that any
real work here.....................12(X + 1/3)^2 + (- 10 - 4/3) = Y

7.  A little arithmetic and,