Completing the Square - A-Level Maths by StudyWell

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:

Take $\alpha$ to be half of $a$ .
Expand $(x+\alpha)^2$ .
Choose $\beta$ so as to adjust the constant so that the original quadratic expression is obtained.

See Example 1.

For the more complicated case where the coefficient of $x^2$ is not 1, remove a factor of $a$ from the original quadratic and perform the above on the inside of the brackets before expanding again in the final step. See Example 2.

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 $y=x^2$ .

Consider the graph of $y=x^2$
The graph of say $y=(x+1)^2$ can be sketched by shifting the graph of $y=x^2$ to the left by 1. See x-transformations on the Transformations page.
The graph of $y=3(x+1)^2$ can be sketched 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.
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.