JEE Advanced Calculus problem

Determine $f(x) = ax^2+bx+c$ , where $a > 0$ , whose roots are $\phi_1$ and $\phi_2$ where $0 < \phi_2 < \phi_1$

If, for some real value $\alpha < \beta$ where $\alpha+\beta = 5$ , $\displaystyle\int_{0}^{\alpha}f(x)dx = 0$ and $\int_{0}^{\phi_1}f(x)dx+\int_{\phi_2}^{\beta}f(x)dx=|\int_{\phi_1}^{\phi_2}f(x)dx|$ The value of $x$ such that $f(x)$ is minimum can be written in the form of $\displaystyle\frac{p}{q}$ , where $p$ and $q$ are coprime. Find $p+q$

Original Question -Also note that alpha is less than beta.

• The answer will be in form of $\frac{p}{q}$ .

• Find p+q.