Prove the quadratic formula for ax^2 + bx + c = 0, where a is non 0 and a,b and c are reals.

By completing the square: ax^2 + bx + c = 0 => x^2 + (bx)/a + c/a = 0 (divide both side by a, since a is non-zero) => (x + b/(2a))^2 + c/a - (b/(2a))^2 = 0 (If this is not immediately clear, try expanding it to obtain line above) => (x + b/(2a))^2 = (b^2 - 4ac)/(2a)^2 => x+ b/(2a) = ±(b^2 - 4ac)^(1/2)/(2a) (square root both side introduce ± signs) => x = (-b ± (b^2 - 4ac)^(1/2))/(2a)