Completing the square is a technique which can help you find turning points of quadratic graphs, and topic=2/11solve quadratic equations/topic.

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(x2+\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{b2}{4a}\Big)+c)}

The completed square form: ((x+p)^2+q)

To complete the square:

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

(ax^{2+bx+c=a\Big(x+\dfrac{b}{2a}\Big)2-\Big(\dfrac{b2}{4a}\Big)+c)}