# Completing the Square – solve or sketch a quadratic

## What is Completing the Square?

**Completing the Square** is when either:

- we write in the form
- or we write in the form

For the simpler case where the coefficient of is 1:

- Firstly, set as half of .
- Secondly, expand .
- Finally, choose so as to adjust the constant so that the original quadratic expression is obtained.

See Example 1. In contrast, if the coefficient of is not 1, first of all, remove a factor of 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 :

- Firstly, consider the graph of . See the Board Example.
- Secondly, sketch the graph of say by shifting the graph of to the left by 1. See -transformations on the Transformations page.
- Thirdly, sketch the graph of by stretching the graph of in the -direction by a factor of 4. See -transformations on the Transformations page.
- Finally, sketching the graph of is done by shifting the graph of down by 1. See -transformations again on the Transformations page.

## Examples

## Videos

Completing the square with unknowns in the coefficients.