Quadratic formula
The quadratic formula is a general expression for the solutions to a quadratic equation. It is used when other methods, such as completing the square, factoring, and square root property do not work or are too tedious.
General Solution For A Quadratic by Completing the Square
Let the quadratic be in the form 
. Moving 
to the other side, we obtain
![\[ax^2+bx=-c.\]](http://latex.artofproblemsolving.com/0/4/1/041b2209b9053f3faf63950f99777465d5de7348.png)
Dividing by 
and adding 
to both sides yields
![\[x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}.\]](http://latex.artofproblemsolving.com/d/c/f/dcff6e02317cebc6ecfba4af26a9bdaa2473b595.png)
Completing the square on the left-hand side gives
![\[\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}.\]](http://latex.artofproblemsolving.com/e/a/6/ea661460f187b78a121c7194fecb67b6a58c6b21.png)
As described above, an equation in this form can be solved, yielding
![\[{x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}}.\]](http://latex.artofproblemsolving.com/e/1/6/e168760868815cd605b465c153971d0fcdad5dff.png)
This formula is also called the quadratic formula. Given the values 
we can find all real and complex solutions to the quadratic equation.
Variation
In some situations, it is preferable to use this variation of the quadratic formula:
![\[\frac{2c}{-b\pm\sqrt{b^2-4ac}}\]](http://latex.artofproblemsolving.com/c/d/5/cd51bd174aea25b1f9d5f1ff18ab4e92e43cd62b.png)
