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.
Contents
General Solution For A Quadratic by Completing the Square
We start with
![\[ax^{2}+bx+c=0\]](http://latex.artofproblemsolving.com/a/b/3/ab3fe06506fbbb606218bea50b951018de7c5f1f.png)
Divide by 
:
![\[x^{2}+\frac{b}{a}x+\frac{c}{a}=0\]](http://latex.artofproblemsolving.com/6/5/4/6544ac4e4bb1bdef8661a062d878fdc02927c757.png)
Add 
to both sides in order to complete the square:
![\[\left(x^{2}+\frac{b}{a}x+\frac{b^{2}}{4a^{2}}\right)+\frac{c}{a}=\frac{b^{2}}{4a^{2}}\]](http://latex.artofproblemsolving.com/3/f/0/3f076f2df75291ea29c243ddce3e9093851988f6.png)
Complete the square:
![\[\left(x+\frac{b}{2a}\right)^{2}+\frac{c}{a}=\frac{b^{2}}{4a^{2}}\]](http://latex.artofproblemsolving.com/c/e/7/ce72b89b26bd1f6a5817ac55314d426b37417558.png)
Move 
to the other side:
![\[\left(x+\frac{b}{2a}\right)^{2}=\frac{b^{2}}{4a^{2}}-\frac{c}{a}=\frac{ab^{2}-4a^{2}c}{4a^{3}}=\frac{b^{2}-4ac}{4a^{2}}\]](http://latex.artofproblemsolving.com/a/1/7/a1714f843e2f925c3a3da3fe38d1ff612e9875d7.png)
Take the square root of both sides:
![\[x+\frac{b}{2a}=\pm\sqrt{\frac{b^{2}-4ac}{4a^{2}}}=\frac{\pm\sqrt{b^{2}-4ac}}{2a}\]](http://latex.artofproblemsolving.com/e/0/2/e024a4ab0baa860faa62c1c8e8c1b0728d6e0d02.png)
Finally, move the 
to the other side:
![\[x=-\frac{b}{2a}+\frac{\pm\sqrt{b^{2}-4ac}}{2a}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\]](http://latex.artofproblemsolving.com/7/d/a/7da525acc5a1d3ef94722ad69fe19c0619051f6c.png)
This is the quadratic formula, and we are done.
Video Solution
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)
