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