The Quadratic Formula
Here we apply the method for completing the square to the general quadratic equation to get a formula
for the solutions to any quadratic equation.
Developing the Formula
Start with the general form of the quadratic equation,
ax^{2} + bx + c = 0.
Assume a is positive for now, and divide each side by a:




= 0 
Subtract
from each side. 



Onehalf of
is
, and
squared is
:
Factor the lefthand side and get a common denominator for the righthand side:
We assumed a was positive so that
would be correct. If a is negative,
then
, and we get
However, the negative sign can be omitted in 2a because of the
Â± symbol preceding
it. For example, the results of 5 Â± (3) and 5
Â± 3 are the same. So when a
is negative, we get the same formula as when a is positive. It is called the quadratic formula.
The Quadratic Formula
The solution to ax^{2} + bx + c = 0, with a
≠ 0, is given by the formula
