When a Quadratic equation does not have real roots

For any equation as $ax^2 + bx + c = 0$ we have $b^2 - 4ac$ or $\Delta$ smaller than zero thus having no real roots if:

1. $a, c > |b|$
2. $a = b = c$
3. $a = b$ and $c > b$ for any positive $b$. You can swap $a$ and $c$.

I have proved these but since it’s trivial, I did not write them here.