What is completing the square?

Completing the square is a method for rewriting a quadratic expression, and can be useful in identifying the minimum value of the expression. The aim is to end up with an expression which looks like c((x+a)^2 + b)Say for example we have the expression y = 4x^2 + 3x + 5. The first thing we need to do it to make the coefficient in front of the x^2 equal to 1 - this makes completing the square a lot easier. To do this we simply factor out the 4:y = 4x^2 + 3x + 5 = 4(x^2 + 3/5 x + 5/4)The next thing to do to make this look like the required form is to half the coefficient in front of the x (the reasoning for this will hopefully become clear later on), so half of 3/5 is 3/10 and we rewrite the expression in a new form. When we do this, we need to take away 9/100, because (x+3/10)^2 = x^2 + 3/5 x + 9/100, but we only want the x^2 + 3/5 x. We also have to add the 5/4 from before. So,y = 4x^2 + 3x + 5 = 4(x^2 + 3/5 x + 5/4) = 4[ (x+3/10)^2 -9/100 + 5/4] = 4[(x + 3/10)^2 -9/100 + 120/100] = 4[(x+3/10)^2 + 111/100]So, y = 4[(x+3/10)^2 + 111/100]And we have completed the square! We can see from this that the minimum value of y will be when (x+3/10) = 0, so x= -3/10, which gives y = 4*111/100 = 111/25.