## What is ‘Completing the Square’?

Completing the square is when either of the following is performed:

• $x^2+ax+b$ is written in the form $(x+\alpha)^2+\beta$
• $ax^2+bx+c$ is written in the form $\alpha(x+\beta)^2+\gamma$

For the simpler case where the coefficient of $x^2$ is 1:

1. Firstly, take $\alpha$ to be half of $a$.
2. Secondly, expand $(x+\alpha)^2$.
3. Finally, choose $\beta$ so as to adjust the constant so that the original quadratic expression is obtained.

See Example 1.

In contrast, the coefficient of $x^2$ may not be 1. First of all, remove a factor of $a$ from the original quadratic. Then perform the above on the inside of the brackets before expanding again in the final step. See Example 2.

### Example 1

Write $x^2+4x+9$ in form $(x+\alpha)^2+\beta$.

First of all, take the coefficient of $x$ in the original quadratic (this is 4) and halve it. See what happens when you choose $\alpha$ to be this value and expand $(x+\alpha)^2$: $(x+2)^2=(x+2)(x+2)=x^2+4x+4$

Now we can see why we should halve the number as you end up with two lots of it in the expansion.
It follows that the result is $x^2+4x+4$ but we want $x^2+4x+9$ and so we must add 5 to this to get $x^2+4x+9$, i.e. choose $\beta$ to be 5. We now have: $x^2+4x+9=(x+2)^2+5$

### 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 then completing the square of what’s inside the brackets: $2x^2+8x-5=2\left(x^2+4x-2.5\right)=2\left((x+2)^2-6.5\right)$

It follows that the final term can be expanded to obtain the result as required: $2x^2+8x-5=2(x+2)^2-13$

Hence, we can see from this that p=q=2 and r=-13.

The graph of a quadratic can easily be sketched if you think about the transformations that have been applied to the graph of $y=x^2$.

• Firstly, consider the graph of $y=x^2$.
• Secondly, sketch the graph of say $y=(x+1)^2$ by shifting the graph of $y=x^2$ to the left by 1. See x-transformations on the Transformations page.
• Thirdly, sketch the graph of $y=3(x+1)^2$ by stretching the graph of $y=(x+1)^2$ about the x-axis by a factor of 3. See y-transformations on the Transformations page.
• Finally, the graph of $y=3(x+1)^2-4$ can then be sketched by shifting the graph of $y=3(x+1)^2$ down by 4. See y-transformations again on the Transformations page.

### Example on the Board – completing the square to sketch a quadratic Click here to find Questions by Topic all scroll down to all past QUADRATIC questions to practice some more Completing the Square questions. Are you ready to test your Pure Maths knowledge? If so, visit our Practice Papers page and take StudyWell’s own Pure Maths tests. Alternatively, try the