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.


Sketching Quadratics

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

Completing the Square Board Question


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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