Question

what is completing the square method

Answer 1

it's a method to solve a quadratic equation....

Answer 2

Say we have a simple expression like x2 + bx. Having x twice in the same expression can make life hard. What can we do?

Well, with a little inspiration from Geometry we can convert it, like this:

Completing the Square Geometry

As you can see x2 + bx can be rearranged nearly into a square ...

... and we can complete the square with (b/2)2

In Algebra it looks like this:

x2 + bx + (b/2)2 = (x+b/2)2
"Complete the Square"
So, by adding (b/2)2 we can complete the square.

And (x+b/2)2 has x only once, which is easier to use.

Keeping the Balance
Now ... we can't just add (b/2)2 without also subtracting it too! Otherwise the whole value changes.

So let's see how to do it properly with an example:

Start with: x^2 + 6x + 7
("b" is 6 in this case)

Complete the Square:


x^2 + 6x + (6/2)^2 + 7 - (6/2)^2
Also subtract the new term

Simplify it and we are done.

simplifies to (x+3)^2
The result:

x2 + 6x + 7 = (x+3)2 − 2

And now x only appears once, and our job is done!

A Shortcut Approach
Here is a quick way to get an answer you may like.

First think about the result we want: (x+d)2 + e

After expanding (x+d)2 that becomes: x2 + 2dx + d2 + e

Now see if we can turn our example into x2 + 2dx + d2 + e form to discover d and e

Example: try to fit x2 + 6x + 7 into x2 + 2dx + d2 + e
x^2 + (6x) + [7] matches x^2 + (2dx) + [d^2+e]

Now we can "force" an answer:

We know that 6x must end up as 2dx, so d must be 3
Next we see that 7 must become d2 + e = 9 + e, so e must be −2
And we get the same result (x+3)2 − 2 as above!



Now, let us look at a useful application: solving Quadratic Equations ...