## Completing the Square

Completing the square is when a quadratic of the form $ax^2+bx+c$ is rewritten in the form $\alpha(x+\beta)^2+\gamma$.

Let us consider the simpler case where $a=1$.

Example 1: Write $x^2+4x+9$ in form $(x+a)^2+b$.
Take the coefficient of $x$ in the original quadratic (this is 4) and halve it – see what happens when you choose $a$ to be this value and expand $(x+a)^2$:

$(x+2)^2=(x+2)(x+2)=x^2+2x+2x+4=x^2+4x+4$

Now we can see why we should halve the number as you end up with two lots of it in the expansion.
The result is $x^2+4x+4$ but we want $x^2+4x+9$ and so we must add 5 to this to get $x^2+4x+9$. It follows that:

$x^2+4x+9=(x+2)^2+5$

We can also think of $x^2+4x$ as $(x+2)^2-4$ and adding 9 to both obtains the required result.

We now consider an example of completing the square where $a\ne 1$.

Example 2 – Write $2x^2+8x-5$ in the form $p(x+q)^2+r$.

Students often get confused with this more complicated example. It can be made simpler to first taking out a factor of 2 and completing the square of what’s inside the brackets:

$2x^2+8x-5=2\left(x^2+4x-2.5\right)=2\left((x+2)^2-4-2.5\right)=2\left((x+2)^2-6.5\right)$

The final term can be expanded as follows to obtain the result as required:

$2x^2+8x-5=2(x+2)^2-13$