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 $$D = 0$$