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.

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