Question 527982
First, divide both sides by a (so that completing the square will be easier).


*[tex \LARGE x^2 + \frac{b}{a}x + \frac{c}{a} = 0]


Now complete the square:


*[tex \LARGE (x + \frac{b}{2a})^2 + C = 0] where C is some unknown value. If you expand and equate the constant terms, you will find that


*[tex \LARGE C = \frac{c}{a} - \frac{b^2}{4a^2} = \frac{4ac - b^2}{4a^2}]


Hence,


*[tex \LARGE (x + \frac{b}{2a})^2 = -C = \frac{b^2 - 4ac}{4a^2}], take square root of both sides.


*[tex \LARGE \pm(x + \frac{b}{2a}) = \frac{\sqrt{b^2 - 4ac}}{2a}]


*[tex \LARGE x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}], the quadratic formula.