Completing the Square – solve or sketch a quadratic

What is Completing the Square?

Completing the Square is when either:

  • we write $x^2+ax+b$ in the form $(x+\alpha)^2+\beta$
  • or we write $ax^2+bx+c$ in the form $\alpha(x+\beta)^2+\gamma$

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

  1. Firstly, set $\alpha$ as 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, if the coefficient of $x^2$ is not 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.

Why is it called Completing the Square?

Sketching Quadratics

Sketching the graph of a quadratic can be easy if you think about the transformations that have been applied to the graph of $y=x^2$:

  • Firstly, consider the graph of $y=x^2$. See the Board Example.
  • 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 $y=4(x+1)^2$ by stretching the graph of $y=(x+1)^2$ in the $y$-direction by a factor of 4. See $y$-transformations on the Transformations page.
  • Finally, sketching the graph of $y=4(x+1)^2-1$ is done by shifting the graph of $y=4(x+1)^2$ down by 1. See $y$-transformations again on the Transformations page.

Questions by Topic

Completing the Square A-Level Questions

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

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