x2 + 14x

To complete a square means to express something into the form (x - a)^2 + a specific number. Putting it in such squared form is the basis of this completing the square.

How nice it would be if we can put them to that form.

Now, I will expand some basic squared terms to illustrate my point.

When we expand the expression (x + 3)^2, we get x2 + 3x + 3x + 3*3 = x2 + 2*3*x + 3^2 = x2 + 6x + 9.

x2 + 6x + 9 is considered to be complete because it can be factorised into (x + 3)^2.

Instead, however, the expression x2 + 6x + 8 does not exhibit the same property because we cannot factorise it into two equivalent brackets. However, if we were to rewrite this as x2 + 6x + 9 - 1, then yes, we recognise the first three terms as (x + 3)^2 and then we have the remaining - 1 attached behind. This is what we call a completed square form.

A general rule of thumb for completing the squares is that for any expression x2 + bx, the term required to complete the brackets is the square of b/2, because in the general expansion of (x + b)^2, the coefficient of x is 2b and the last term is the square of b, or the square of “half of 2b”.

We must return the same term which we have introduced to maintain equality.

So for the first example,

x2 + 14x (and half of 14 is 7)
= x2 + 14x + (14/2)^2 - (14/2)^2
= x2 + 14x + 7^2 - 7^2
= x2 + 14x + 49 - 49
= (x + 7) (x + 7) - 49
= (x + 7)^2 - 49

is our completed square form for x2 + 14. The fourth and fifth lines are not necessary to be written down in your workings.