Discriminant of a quadratic equation

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

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

$$x^2+2x+5=0 \rightarrow D= 2^2-4 \cdot 5= 4-20=-16$$

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

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

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

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 $\mathbb{R}$

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

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