Discriminant of a quadratic equation

The discriminant of a quadratic equation $a x^{2} + b x + c = 0$ is a number, indicated with the letter $D$ (in some texts the Greek letter $Δ$ is used) whose value is calculated as follows: $D = b^{2} - 4 a c$

Example

$x^{2} + 3 x - 10 = 0 \to D = 3^{2} - 4 \cdot 1 \cdot (- 10) = 9 + 40 = 49$

$x^{2} + 2 x + 5 = 0 \to D = 2^{2} - 4 \cdot 5 = 4 - 20 = - 16$

$x^{2} - 16 = 0 \to D = - 4 \cdot 1 \cdot (- 16) = 64$

So the discriminant is the expression underneath the square root in the general solution of the equation.

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

When the discriminant is zero, the equation will have just one solution (it is also said that the equation has a double solution).

If it is less than zero, since there are not square roots of negative numbers, the equation will have no solutions.

$D > 0$ two solutions
$D = 0$ one solution
$D < 0$ no solutions in $R$

Example

In the previous examples we can say, with no need to solve the equations, that:

$x^{2} + 3 x - 10 = 0$ has two solutions, since $D = 49 > 0$
$x^{2} + 2 x + 5 = 0$ has no solutions, since $D = - 16 < 0$
$x^{2} - 4 x + 4 = 0$ has one solution, since $D = 0$