You should recall a method called completing the square from your studies of solving quadratic equations.
|
To refresh it in your minds we review the algebraic method of completing the square.
|
How can we think about completing the square algebraically?
|
Given an expression of the form ax2 + bx, you want to determine the appropriate number to add to the expression so that it will become an expression of the form a(x + p)2, where p is a real number that depends on a and b in a way we will figure out.
|
It is much easier to follow the calculations when a = 1. So let's see how to create (x + p)2 out of x2 + bx.
|
Working backwords is often a good strategy, especially if you do it off the record on scrap paper. People think you are a genius when you pull the right number out of the air.
- Start with the target (x + p)2 and expand it
out to get x2 + 2px +
p2.
- Notice the coefficient of x is 2p.
- In our
starting expression, the coefficient of x is b.
- So the only way things could work out is if we used p = b/2.
- We need to add in the p2 =
b2/4 to "complete the square".
|
|
Remember, anything you add into the expression has to be subtracted as well so you don't change the total value.
|
Here is the calculation you show. Everyone will be impressed with your stroke of genius at the first step.
x2 + bx | = |
x2 + bx + b2/4 -
b2/4 |
| = |
(x2 + bx + b2/4) - b2/4 |
| = |
(x + b/2)2 -
b2/4 |
|
|