What is completing the square?
What is completing the square?Completing the square is a way of rewriting quadratic expressions in a form that can help us interpret graphs and solve quadratic equations. It involves rewriting ax2 + bx + c in the form (x + d)2 + e, where d and e are new constants to be found.
How do I do it?Let's take the simple quadratic equation x2 + 2x + 1 as an example. The goal is to write it in the form (x + d)2 + e, but how do we find d and e? There are several ways of doing it, but here is a quick way you can use as a shortcut.First, take the number in front of x, (called the coefficient of x) and divide it by 2. This goes inside the bracket next to x. So for x2 + 2x + 1, the coefficient of x is 2. 2 divided by 2 is 1, so we would write it down like this: (x + 1)2.The next step is to subtract the square of the number we just put in the bracket next to x. 12 = 1, so we would write: (x + 1)2 - 1.Finally, we add on the number at the end of the original quadratic equation (the constant term). The number at the end of x2 + 2x + 1 is 1. We write (x + 1)2 - 1 + 1. The lines of working would look like this:x2 + 2x + 1= (x + 1)2 - 1 + 1= (x + 1)2
What about when a ≠ 0?The case is slightly different when the coefficient of x2 is not 0. In that case we do the following method. Taking the example of 3x2 + 6x - 9, we must first factor out the coefficient of x2 like so: 3x2 + 6x - 9 = 3[x2 + 2x - 3]From here, we proceed as normal:3[x2 + 2x - 3]= 3[(x+1)2 - 1 + 3]= 3[(x+1)2 + 2]= 3(x+1)2 + 6
