how to form the quadratic function into y=ax^2+bx+c

how to form the quadratic function into y=ax^2+bx+c  ?

$$\begin{array}{rcl} y&=& ax^2+bx+c\\\\ y&=& a\left( x^2+{b\over a}x\right) +c\\\\ y&=& a\left( x^2+{b\over a}x \textcolor[rgb]{1,0,0}{ + ({b\over 2a})^2 - ({b\over 2a})^2 } \right) +c\\\\ y&=& a\left[(x+{b\over 2a})^2 - ({b\over 2a})^2 \right] +c\\\\ y&=&a (x+{b\over 2a})^2 - {b^2\over 4a}+c\\\\ y&=&a (x+{b\over 2a})^2 + c - {b^2\over 4a} \end{array}$$

Morgan Tud contributed this thread a while back.

http://web2.0calc.com/questions/magick-of-the-quadratic#rr2