Where does the quadratic formulae come from?

The general form of a quadratic equation is, ax² +bx + c = 0. If we divide all terms by a we get, x² +(b/a)x + c/a = 0. then by completing the square we get (x+(b/2a))² + c/a - b²/4a² = 0 which rearanged is, (x+(b/2a))² = b²/4a² -  c/a. We can combine the 2 terms on the left hand side of this equation into, b²-4ac which gives the overall equation,

(x+(b/2a))² = (b²-4ac) / 4a². If we then square root both sides we have x+b/2a = sqrt(b²-4ac)/2a. By rearanging again we get x= (sqrt(b²-4ac)-b) / 2a. Which looks like the quadratic formulae, there is a subtlety which is that a square root can be either + or -, i.e. the sqrt(4) is 2 or -2 therefore for our quadratic formulae we have to consider both the postivie and negative terms.

Therefore our overall result reduces to the formulae you are familiar with x= (-b ± sqrt(b²-4ac)) / 2a