Quadratic formula

For the quadratic equation $$ax^2+bx+c=0$$, the real solutions are given by the quadratic formula:


 * $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Discriminant
The discriminant $$\Delta$$ in the quadratic equation is given by:


 * $$\Delta=b^2-4ac$$

The discriminant can be used to provide information about the number of solutions to a quadratic equation:


 * if $$\Delta>0$$, there are two unique real solutions
 * if $$\Delta=0$$, there is one unique real solution
 * if $$\Delta<0$$, there are no unique real solutions.

Example
Suppose one needed to find solutions for the equation $$x^2-6=x$$.

First we rearrange the equation to make it equal zero:


 * $$x^2-6=x$$


 * $$\therefore x^2-x-6=0$$

Now, using the quadratic formula:


 * $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$


 * $$a=1, b=-1, c=-6 \therefore x=\frac{-(-1)\pm\sqrt{(-1)^2-4(1)(-6)}}{2(1)}$$


 * $$\therefore x=\frac{1\pm\sqrt{25}}{2}=\frac{1\pm5}{2}$$


 * $$\therefore$$ $$x=\frac{6}{2}=3$$ or $$x=\frac{-4}{2}=-2$$.