Completing the square is a technique which can help you find turning points of quadratic graphs, and solve quadratic equations.

The completed square form looks like this: \((x+p)^2+q\)

Completing the square is simple for quadratic functions where \(a=1\):

\(x^2+bx+c=\Big(x+\dfrac{b}{2}\Big)^2-\Big(\dfrac{b}{2}\Big)^2+c\)

It is more complicated for quadratic functions where \(a>1\), because you need to factorise out \(a\) before completing the square and simplifying:

\(ax^2+bx+c=a\Big(x^2+\dfrac{b}{a}x\Big)+c\)

\(\qquad\qquad=a\Big[\Big(x+\dfrac{b}{2a}\Big)^2-\Big(\dfrac{b}{2a}\Big)^2\Big]+c\)

\(\qquad\qquad=a\Big(x+\dfrac{b}{2a}\Big)^2-\Big(\dfrac{b^2}{4a}\Big)+c\)