Quadratic equation calculator

A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. Enter the coefficients a, b and c into the calculator and it works out the discriminant and the roots of the equation. If a term is missing, write a zero in its place: the equation x² − 9 = 0 has a = 1, b = 0 and c = −9. Be careful with signs: from x² − 5x + 6 = 0, enter b = −5, not 5.

How the calculator works

First it works out the discriminant:

$$D=b^2-4ac$$

Its sign then tells it how many solutions the equation has. A positive discriminant gives two different roots, a zero discriminant gives one, and a negative discriminant means there are no real solutions. The roots are

$$x_{1,2}=\frac{-b\pm \sqrt{D}}{2a}$$

For the default equation x² + 7x + 12 = 0, D = 7² − 4 · 1 · 12 = 1, so

$$x_{1,2}=\frac{-7\pm 1}{2},x_1=-3,x_2=-4$$

The calculator also copes with a negative discriminant: it takes the square root of a negative number in complex numbers. A fuller explanation, along with simpler methods for equations missing b or c, is in the article on quadratic equations.