What is completing the square and how do I do it?

Completing the square is a method of solving quadratic equations that can't be factorised. The end goal is to express the quadratic in the form:(x + a)² = -bWhich allows us to root both sides and find our solutions for x.Let's do an example. We'll complete the square for the quadratic:x² + 8x + 4 = 0First, we group the x terms together in a bracket to make the process a little easier to visualise:(x² + 8x) + 4 = 0We're then going to rewrite (x² + 8x) as (x + 4)² - 4². We've done this by moving the square to the outside of the bracket, removing the x from the second term, halving the second term's coefficient, and squaring this number and then taking it away from the outside of the bracket. We can see that these are equal expressions by expanding them out:x² + 8x = (x + 4)(x + 4) - 4² = x² + 8x + 16 -16x² + 8x = x² + 8xPutting this back into the equation we started with, we get:(x + 4)² + 4 - 16 = 0(x + 4)² - 12 = 0(x + 4)² = 12In order to find our solutions for x, we now root both sides, remembering that the root of a positive number gives us two answers: one negative and one positive:x + 4 = ±√(12)So, x = -4 ±√(12)