1. The nature of roots of a quadratic equation $ax^2+bx+c=0$ depends on the discriminant $\Delta=b^2-4ac$. 2. If $\Delta>0$, there are two distinct real roots. Example: $x^2-5x+6=0$ with $\Delta=25-24=1>0$, roots are $x=2$ and $x=3$. 3. If $\Delta=0$, there is exactly one real root (a repeated root). Example: $x^2-4x+4=0$ with $\Delta=16-16=0$, root is $x=2$. 4. If $\Delta<0$, there are two complex conjugate roots. Example: $x^2+4x+5=0$ with $\Delta=16-20=-4<0$, roots are $x=-2+i$ and $x=-2 - i$. These examples show the three possible natures of roots: distinct real, repeated real, and complex conjugates.