Quadratic formula

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For the quadratic equation $a x 2 + b x + c = 0$ , the real solutions are given by the quadratic formula:

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

[edit] Discriminant

The discriminant $Δ$ in the quadratic equation is given by:

Δ = b 2 - 4 a c

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

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 $x=\frac{-4}{2}=-2$ .