# A Real Equation!

Algebra Level 5

If $\displaystyle f(x) = a_1x^3 + a_2x^2 + a_3 = 0 \text{ where } a_1, a_2 \in \R^+ \text{ and } a_3 \in \R$ has three distinct real roots, then the exhaustive range of $\displaystyle a_3$ is $\displaystyle a_3 \in \bigg(-\frac{ba_2^c}{da_1^e}, -f\bigg) \text{ where } b, c, d, e, f \in I \text{ and } \gcd( b, d) = 1$, enter answer as $b + c + d + e + f$.

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