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:
- Firstly, set $\alpha$ as half of $a$.
- Secondly, expand $(x+\alpha)^2$.
- 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.
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.