The quadratic formula,
can be derived by on the general standard form a quadratic equation. Recall that completing the square is a method for solving quadratic equations. Thus, completing the square on the general form of a quadratic equation,
ax2+bx+c=0,
creates another way to solve all quadratic equations. By completing the square, it's possible to isolate
x. The first step is to rewrite the equation by moving
c and
a to the right-hand side.
ax2+bx+c=0
ax2+bx=-c
aax2+bx=-ac
aax2+abx=-ac
x2+abx=-ac
x2+ab⋅x=-ac
Next, a new constant term can be added to both sides of the equation, so the expression on the left becomes a perfect square trinomial. The coefficient of the
x-term is
ab. Thus, the term needed to complete the square is
(ab/2)2. For equality to hold, this term is added on both sides.
x2+ab⋅x=-ac
x2+ab⋅x+(ab/2)2=(ab/2)2−ac
x2+ab⋅x+(2ab)2=(2ab)2−ac
Now that the left-hand side is a perfect square trinomial, it can be written in factored form. The right-hand side can also be simplified.
x2+ab⋅x+(2ab)2=(2ab)2−ac
(x+2ab)2=(2ab)2−ac
(x+2ab)2=(2a)2b2−ac
(x+2ab)2=4a2b2−ac
(x+2ab)2=4a2b2−4a24ac
(x+2ab)2=4a2b2−4ac
Now, there is one
x-term. To isolate
x, it is necessary to square-root both sides of the equation. This results in both a positive and a negative term on the right-hand side.
Thus, it is possible to solve any quadratic equation in standard form using