Quadratic Properties Required

Let's start with the monic quadratic in the form: $x^2+bx+c=0$ . Here, they say if $x=0$ , then the quadratic equation equal 30.

$0^2+0+c=30\Rightarrow c=30$ .

It is also stated that inputting 2 as $x$ will make the quadratic equation equal 0. In other words:

$2^2+2b+30=0\Rightarrow4+2b=-30$

$\Rightarrow b=-17$

From this, we can rewrite the equation as: $x^2-17x+30=0$

By Vieta's formula, the sum of the roots will be

$-\dfrac{b}{a}=-\dfrac{-17}{1}=\boxed{17}$

Let the quadratic equation be $ax^{2}+bx+c=0$ $f(0)=30$ $\implies$ Constant term of the quadratic equation is 30 . Also, $f(2)=0$ $\implies$ $x-2$ is the factor of the equation. Now, in order for 30 to be the constant the other factor must be $x-15$ making the equation $(x-2)(x-15)=x^2-17x+30$ , $\therefore$ sum of roots is equal to $\boxed{17}$