If $a,b\in R$ and $a \neq 0$ and quadratic $ax^2-bx+c=0$ If $a,b\in R$ and $a \neq 0$ and quadratic $ax^2-bx+c=0$ has imaginary roots  then  $a+b+1$
why a+b+1 should be positive?

The roots of $ax^2-bx+c$ are given by 
$$
\frac{b\pm\sqrt{b^2-4ac}}{2},
$$
which are both complex if and only if $b^2-4ac<0$.
