Completing the Square
Before we can complete the square we need to be able to recognise a perfect square. This has been discussed previously but the next exercise will act as a good refresher and shall help to focus your mind on a very important concept.
Exercise: what should c be in order to have a perfect square?
The following quadratic equation is not a perfect square. Input the number c should be in order for the equation to a perfect square.
You can check the reasoning below:
I couldn't help but admire the simple but clever reasoning behind completing the square when I first encountered it all those years ago.
The completing the square procedure can also be understood visually: press the buttons below to literally see
"completing the square".
Finding the Solutions
The completed square form allows one to find the solutions quite easily, all one has to do is make the bracket the subject and take the square root of both sides. Try the follow exercise several times; practice makes perfect.
Exercise:Find the Solutions by Completing the Square
New or Exercise.
You can check you working out below:
Summary
Completing the square is a powerful method of finding the solutions of a quadratic equation. The examples and questions in this section all have integer solution, however completing the square can be applied to quadratic equations where the solutions can be fractions or non-integers, or may not even exist.