Complete the Square

Questions and worked solutions at the bottom of the page.

Completing the square is a method to solve quadratic equations that is useful when factorising is not possible (or too difficult). We will complete the square for a generic quadratic;

ax² + bx + c = 0

Then you can use this to solve all completing the square questions by substituting in the right values for a, b and c depending on the question

For example if we were given the equation 3x² – 4x + 5 = 0 then a = 3, b = -4, and c = 5.

So lets start from

ax² + bx + c = 0

And that’s the quadratic formula! Coincidence? I don’t think so. This is where the quadratic formula comes from but rest easy you do not have to be able to derive the quadratic formula for your exam. That being said if you can do that you can do any complete the square question they throw at you.

It looks worse than it really is when we are using letters, it makes the algebra worse. When we are doing a proper question with numerical values for a, b and c it’s a lot easier, plus for a lot of questions a=1 which makes thing much simpler.