Quadratic Equation - Prove the following

Question

If $c$ is positive and $2ax^2 + 3bx + 5c$ does not have any real roots, then prove that $2a - 3b + 5c > 0$.

Answer 1

If $c$ is positive, then you get that the polynomial $2ax^2+3bx+5c$ has a stricly positive value for $x=0$.

Since it is continuous and does not have real roots, you know that it will always be strictly positive (for all $x$).

You can then take $x=-1$, and you get

$$2a-3b+5c>0.$$