ax^2 + bx +c = 0 Divide all terms by a so as to reduce the coefficient of x^2 to 1 x^2 +b/a x + c/a=0 Subtract the constant term from both sides of the equation x^2+b/a x = - c/a To have a square on the left side the third term (constant) should be (b/(2a))^2 So add that amount to both sides x^2 +b/a x +(b/(2a))^2 = (b/(2a))^2 - c/a Re-write the left-side as a square (x+(b/(2a)))^2 = (b/(2a))^2 -c/a Take the square root of both sides (remembering that the result could be plus or minus) x+(b/(2a)) = +-sqrt((b/(2a))^2 -c/a) Subtract the constant term on the left side from both sides x = +-sqrt((b/(2a))^2 -c/a) - (b/(2a)) or, with some simplification x= (-b+-sqrt(b^2-4ac))/(2a) (the standard form for solving a quadratic)

Completing the square is a useful technique for solving quadratic equations. It is a more powerful technique than factorisation because it can be applied to equations that do not factorise.

The Quadratic Formula