Need help solving difficult equation

Question

Need help solving for $X_{\text{ave}}$:

$$X_{\text{ave}} = \left(\frac{a_1 f_1^2} {F_0 + F_r\, f_{\text{carb}}\, f_{\text{calc}} \, (1-f_1)} +\frac{a_2 f_2^2} {F_0 + F_r \, f_{\text{carb}} \, f_{\text{calc}} \, (1-f_2)} + \frac{b}{F_0} \right) (F_0 + F_r r_0) \, f_{\text{calc}} $$ where $$f_{\text{carb}} = \frac{X_{\text{carb}} - X_{\text{calc}}} {X_{\text{ave}} - X_{\text{calc}}}$$ and $$f_{\text{calc}} = \frac{X_{\text{carb}} - X_{\text{calc}}} {X_{\text{carb}}}\,.$$

Answer 1

MATLAB returns

F0^3*Xcarb^3*z^3 + (- 2*F0^3*Xcalc*Xcarb^3 + a1*F0^3*Xcalc*Xcarb^2*f1^2 + a2*F0^3*Xcalc*Xcarb^2*f2^2 + b*F0^3*Xcalc*Xcarb^2 - a1*F0^3*Xcarb^3*f1^2 - a2*F0^3*Xcarb^3*f2^2 - b*F0^3*Xcarb^3 - Fr*F0^2*Xcalc^2*Xcarb^2*f1 - Fr*F0^2*Xcalc^2*Xcarb^2*f2 + 2*Fr*F0^2*Xcalc^2*Xcarb^2 + 2*Fr*F0^2*Xcalc*Xcarb^3*f1 + 2*Fr*F0^2*Xcalc*Xcarb^3*f2 - 4*Fr*F0^2*Xcalc*Xcarb^3 + Fr*a1*r0*F0^2*Xcalc*Xcarb^2*f1^2 + Fr*a2*r0*F0^2*Xcalc*Xcarb^2*f2^2 + Fr*b*r0*F0^2*Xcalc*Xcarb^2 - Fr*F0^2*Xcarb^4*f1 - Fr*F0^2*Xcarb^4*f2 + 2*Fr*F0^2*Xcarb^4 - Fr*a1*r0*F0^2*Xcarb^3*f1^2 - Fr*a2*r0*F0^2*Xcarb^3*f2^2 - Fr*b*r0*F0^2*Xcarb^3)*z^2 + (F0*Fr^2*Xcarb^5 + F0^3*Xcalc^2*Xcarb^3 - 4*F0*Fr^2*Xcalc*Xcarb^4 + F0*Fr^2*Xcalc^4*Xcarb - 2*F0^2*Fr*Xcalc*Xcarb^4 - 2*F0^2*Fr*Xcarb^4*b - F0*Fr^2*Xcarb^5*f1 - F0*Fr^2*Xcarb^5*f2 + 2*F0^3*Xcalc*Xcarb^3*b + 6*F0*Fr^2*Xcalc^2*Xcarb^3 - 4*F0*Fr^2*Xcalc^3*Xcarb^2 + 4*F0^2*Fr*Xcalc^2*Xcarb^3 - 2*F0^2*Fr*Xcalc^3*Xcarb^2 - 2*F0^3*Xcalc^2*Xcarb^2*b - 6*F0^2*Fr*Xcalc^2*Xcarb^2*b - 6*F0*Fr^2*Xcalc^2*Xcarb^3*f1 + 4*F0*Fr^2*Xcalc^3*Xcarb^2*f1 - 2*F0^2*Fr*Xcalc^2*Xcarb^3*f1 + F0^2*Fr*Xcalc^3*Xcarb^2*f1 - 6*F0*Fr^2*Xcalc^2*Xcarb^3*f2 + 4*F0*Fr^2*Xcalc^3*Xcarb^2*f2 - 2*F0^2*Fr*Xcalc^2*Xcarb^3*f2 + F0^2*Fr*Xcalc^3*Xcarb^2*f2 - F0^2*Fr*Xcarb^4*a1*f1^2 - F0^2*Fr*Xcarb^4*a2*f2^2 + 2*F0^3*Xcalc*Xcarb^3*a1*f1^2 + 2*F0^3*Xcalc*Xcarb^3*a2*f2^2 - 2*F0^3*Xcalc^2*Xcarb^2*a1*f1^2 - 2*F0^3*Xcalc^2*Xcarb^2*a2*f2^2 + 6*F0^2*Fr*Xcalc*Xcarb^3*b + 2*F0^2*Fr*Xcalc^3*Xcarb*b + 4*F0*Fr^2*Xcalc*Xcarb^4*f1 - F0*Fr^2*Xcalc^4*Xcarb*f1 + F0^2*Fr*Xcalc*Xcarb^4*f1 + 4*F0*Fr^2*Xcalc*Xcarb^4*f2 - F0*Fr^2*Xcalc^4*Xcarb*f2 + F0^2*Fr*Xcalc*Xcarb^4*f2 + F0^2*Fr*Xcarb^4*b*f1 + F0^2*Fr*Xcarb^4*b*f2 + F0*Fr^2*Xcarb^5*f1*f2 - 2*F0*Fr^2*Xcarb^4*b*r0 - 3*F0^2*Fr*Xcalc*Xcarb^3*b*f1 - F0^2*Fr*Xcalc^3*Xcarb*b*f1 - 3*F0^2*Fr*Xcalc*Xcarb^3*b*f2 - F0^2*Fr*Xcalc^3*Xcarb*b*f2 - 4*F0*Fr^2*Xcalc*Xcarb^4*f1*f2 + F0*Fr^2*Xcalc^4*Xcarb*f1*f2 + 6*F0*Fr^2*Xcalc*Xcarb^3*b*r0 + 2*F0*Fr^2*Xcalc^3*Xcarb*b*r0 + 2*F0^2*Fr*Xcalc*Xcarb^3*b*r0 + F0*Fr^2*Xcarb^4*b*f1*r0 + F0*Fr^2*Xcarb^4*b*f2*r0 + 3*F0^2*Fr*Xcalc*Xcarb^3*a1*f1^2 + F0^2*Fr*Xcalc^3*Xcarb*a1*f1^2 + 3*F0^2*Fr*Xcalc*Xcarb^3*a2*f2^2 + F0^2*Fr*Xcalc^3*Xcarb*a2*f2^2 + 3*F0^2*Fr*Xcalc^2*Xcarb^2*b*f1 + 3*F0^2*Fr*Xcalc^2*Xcarb^2*b*f2 + 6*F0*Fr^2*Xcalc^2*Xcarb^3*f1*f2 - 4*F0*Fr^2*Xcalc^3*Xcarb^2*f1*f2 - 6*F0*Fr^2*Xcalc^2*Xcarb^2*b*r0 - 2*F0^2*Fr*Xcalc^2*Xcarb^2*b*r0 + F0^2*Fr*Xcarb^4*a1*f1^2*f2 + F0^2*Fr*Xcarb^4*a2*f1*f2^2 - F0*Fr^2*Xcarb^4*a1*f1^2*r0 - F0*Fr^2*Xcarb^4*a2*f2^2*r0 - 3*F0^2*Fr*Xcalc^2*Xcarb^2*a1*f1^2 - 3*F0^2*Fr*Xcalc^2*Xcarb^2*a2*f2^2 + F0*Fr^2*Xcarb^4*a1*f1^2*f2*r0 + F0*Fr^2*Xcarb^4*a2*f1*f2^2*r0 + 3*F0^2*Fr*Xcalc^2*Xcarb^2*a1*f1^2*f2 + 3*F0^2*Fr*Xcalc^2*Xcarb^2*a2*f1*f2^2 - 3*F0*Fr^2*Xcalc^2*Xcarb^2*a1*f1^2*r0 - 2*F0^2*Fr*Xcalc^2*Xcarb^2*a1*f1^2*r0 - 3*F0*Fr^2*Xcalc^2*Xcarb^2*a2*f2^2*r0 - 2*F0^2*Fr*Xcalc^2*Xcarb^2*a2*f2^2*r0 - 3*F0*Fr^2*Xcalc*Xcarb^3*b*f1*r0 - F0*Fr^2*Xcalc^3*Xcarb*b*f1*r0 - 3*F0*Fr^2*Xcalc*Xcarb^3*b*f2*r0 - F0*Fr^2*Xcalc^3*Xcarb*b*f2*r0 - 3*F0^2*Fr*Xcalc*Xcarb^3*a1*f1^2*f2 - F0^2*Fr*Xcalc^3*Xcarb*a1*f1^2*f2 - 3*F0^2*Fr*Xcalc*Xcarb^3*a2*f1*f2^2 - F0^2*Fr*Xcalc^3*Xcarb*a2*f1*f2^2 + 3*F0*Fr^2*Xcalc*Xcarb^3*a1*f1^2*r0 + F0*Fr^2*Xcalc^3*Xcarb*a1*f1^2*r0 + 2*F0^2*Fr*Xcalc*Xcarb^3*a1*f1^2*r0 + 3*F0*Fr^2*Xcalc*Xcarb^3*a2*f2^2*r0 + F0*Fr^2*Xcalc^3*Xcarb*a2*f2^2*r0 + 2*F0^2*Fr*Xcalc*Xcarb^3*a2*f2^2*r0 + 3*F0*Fr^2*Xcalc^2*Xcarb^2*b*f1*r0 + 3*F0*Fr^2*Xcalc^2*Xcarb^2*b*f2*r0 - 3*F0*Fr^2*Xcalc*Xcarb^3*a1*f1^2*f2*r0 - F0*Fr^2*Xcalc^3*Xcarb*a1*f1^2*f2*r0 - 3*F0*Fr^2*Xcalc*Xcarb^3*a2*f1*f2^2*r0 - F0*Fr^2*Xcalc^3*Xcarb*a2*f1*f2^2*r0 + 3*F0*Fr^2*Xcalc^2*Xcarb^2*a1*f1^2*f2*r0 + 3*F0*Fr^2*Xcalc^2*Xcarb^2*a2*f1*f2^2*r0)*z + F0*Fr^2*Xcalc^5*b - F0*Fr^2*Xcarb^5*b + Fr^3*Xcalc^5*b*r0 - Fr^3*Xcarb^5*b*r0 - F0^3*Xcalc^2*Xcarb^3*b + F0^3*Xcalc^3*Xcarb^2*b - 10*F0*Fr^2*Xcalc^2*Xcarb^3*b + 10*F0*Fr^2*Xcalc^3*Xcarb^2*b - 6*F0^2*Fr*Xcalc^2*Xcarb^3*b + 6*F0^2*Fr*Xcalc^3*Xcarb^2*b - 10*Fr^3*Xcalc^2*Xcarb^3*b*r0 + 10*Fr^3*Xcalc^3*Xcarb^2*b*r0 - F0^3*Xcalc^2*Xcarb^3*a1*f1^2 + F0^3*Xcalc^3*Xcarb^2*a1*f1^2 - F0^3*Xcalc^2*Xcarb^3*a2*f2^2 + F0^3*Xcalc^3*Xcarb^2*a2*f2^2 + 5*F0*Fr^2*Xcalc*Xcarb^4*b - 5*F0*Fr^2*Xcalc^4*Xcarb*b + 2*F0^2*Fr*Xcalc*Xcarb^4*b - 2*F0^2*Fr*Xcalc^4*Xcarb*b - F0*Fr^2*Xcalc^5*b*f1 - F0*Fr^2*Xcalc^5*b*f2 + F0*Fr^2*Xcarb^5*b*f1 + F0*Fr^2*Xcarb^5*b*f2 + 5*Fr^3*Xcalc*Xcarb^4*b*r0 - 5*Fr^3*Xcalc^4*Xcarb*b*r0 - Fr^3*Xcalc^5*b*f1*r0 - Fr^3*Xcalc^5*b*f2*r0 + Fr^3*Xcarb^5*b*f1*r0 + Fr^3*Xcarb^5*b*f2*r0 - 5*F0*Fr^2*Xcalc*Xcarb^4*b*f1 + 5*F0*Fr^2*Xcalc^4*Xcarb*b*f1 - F0^2*Fr*Xcalc*Xcarb^4*b*f1 + F0^2*Fr*Xcalc^4*Xcarb*b*f1 - 5*F0*Fr^2*Xcalc*Xcarb^4*b*f2 + 5*F0*Fr^2*Xcalc^4*Xcarb*b*f2 - F0^2*Fr*Xcalc*Xcarb^4*b*f2 + F0^2*Fr*Xcalc^4*Xcarb*b*f2 + 2*F0*Fr^2*Xcalc*Xcarb^4*b*r0 - 2*F0*Fr^2*Xcalc^4*Xcarb*b*r0 + F0*Fr^2*Xcalc^5*b*f1*f2 - F0*Fr^2*Xcarb^5*b*f1*f2 - 5*Fr^3*Xcalc*Xcarb^4*b*f1*r0 + 5*Fr^3*Xcalc^4*Xcarb*b*f1*r0 - 5*Fr^3*Xcalc*Xcarb^4*b*f2*r0 + 5*Fr^3*Xcalc^4*Xcarb*b*f2*r0 + Fr^3*Xcalc^5*b*f1*f2*r0 - Fr^3*Xcarb^5*b*f1*f2*r0 + F0^2*Fr*Xcalc*Xcarb^4*a1*f1^2 - F0^2*Fr*Xcalc^4*Xcarb*a1*f1^2 + F0^2*Fr*Xcalc*Xcarb^4*a2*f2^2 - F0^2*Fr*Xcalc^4*Xcarb*a2*f2^2 + 10*F0*Fr^2*Xcalc^2*Xcarb^3*b*f1 - 10*F0*Fr^2*Xcalc^3*Xcarb^2*b*f1 + 3*F0^2*Fr*Xcalc^2*Xcarb^3*b*f1 - 3*F0^2*Fr*Xcalc^3*Xcarb^2*b*f1 + 10*F0*Fr^2*Xcalc^2*Xcarb^3*b*f2 - 10*F0*Fr^2*Xcalc^3*Xcarb^2*b*f2 + 3*F0^2*Fr*Xcalc^2*Xcarb^3*b*f2 - 3*F0^2*Fr*Xcalc^3*Xcarb^2*b*f2 - 6*F0*Fr^2*Xcalc^2*Xcarb^3*b*r0 + 6*F0*Fr^2*Xcalc^3*Xcarb^2*b*r0 - F0^2*Fr*Xcalc^2*Xcarb^3*b*r0 + F0^2*Fr*Xcalc^3*Xcarb^2*b*r0 + 10*Fr^3*Xcalc^2*Xcarb^3*b*f1*r0 - 10*Fr^3*Xcalc^3*Xcarb^2*b*f1*r0 + 10*Fr^3*Xcalc^2*Xcarb^3*b*f2*r0 - 10*Fr^3*Xcalc^3*Xcarb^2*b*f2*r0 - 3*F0^2*Fr*Xcalc^2*Xcarb^3*a1*f1^2 + 3*F0^2*Fr*Xcalc^3*Xcarb^2*a1*f1^2 - 3*F0^2*Fr*Xcalc^2*Xcarb^3*a2*f2^2 + 3*F0^2*Fr*Xcalc^3*Xcarb^2*a2*f2^2 - 10*Fr^3*Xcalc^2*Xcarb^3*b*f1*f2*r0 + 10*Fr^3*Xcalc^3*Xcarb^2*b*f1*f2*r0 + 3*F0^2*Fr*Xcalc^2*Xcarb^3*a1*f1^2*f2 - 3*F0^2*Fr*Xcalc^3*Xcarb^2*a1*f1^2*f2 + 3*F0^2*Fr*Xcalc^2*Xcarb^3*a2*f1*f2^2 - 3*F0^2*Fr*Xcalc^3*Xcarb^2*a2*f1*f2^2 - 3*F0*Fr^2*Xcalc^2*Xcarb^3*a1*f1^2*r0 + 3*F0*Fr^2*Xcalc^3*Xcarb^2*a1*f1^2*r0 - F0^2*Fr*Xcalc^2*Xcarb^3*a1*f1^2*r0 + F0^2*Fr*Xcalc^3*Xcarb^2*a1*f1^2*r0 - 3*F0*Fr^2*Xcalc^2*Xcarb^3*a2*f2^2*r0 + 3*F0*Fr^2*Xcalc^3*Xcarb^2*a2*f2^2*r0 - F0^2*Fr*Xcalc^2*Xcarb^3*a2*f2^2*r0 + F0^2*Fr*Xcalc^3*Xcarb^2*a2*f2^2*r0 + 5*F0*Fr^2*Xcalc*Xcarb^4*b*f1*f2 - 5*F0*Fr^2*Xcalc^4*Xcarb*b*f1*f2 - F0*Fr^2*Xcalc*Xcarb^4*b*f1*r0 + F0*Fr^2*Xcalc^4*Xcarb*b*f1*r0 - F0*Fr^2*Xcalc*Xcarb^4*b*f2*r0 + F0*Fr^2*Xcalc^4*Xcarb*b*f2*r0 + 5*Fr^3*Xcalc*Xcarb^4*b*f1*f2*r0 - 5*Fr^3*Xcalc^4*Xcarb*b*f1*f2*r0 - F0^2*Fr*Xcalc*Xcarb^4*a1*f1^2*f2 + F0^2*Fr*Xcalc^4*Xcarb*a1*f1^2*f2 - F0^2*Fr*Xcalc*Xcarb^4*a2*f1*f2^2 + F0^2*Fr*Xcalc^4*Xcarb*a2*f1*f2^2 - 10*F0*Fr^2*Xcalc^2*Xcarb^3*b*f1*f2 + 10*F0*Fr^2*Xcalc^3*Xcarb^2*b*f1*f2 + F0*Fr^2*Xcalc*Xcarb^4*a1*f1^2*r0 - F0*Fr^2*Xcalc^4*Xcarb*a1*f1^2*r0 + F0*Fr^2*Xcalc*Xcarb^4*a2*f2^2*r0 - F0*Fr^2*Xcalc^4*Xcarb*a2*f2^2*r0 + 3*F0*Fr^2*Xcalc^2*Xcarb^3*b*f1*r0 - 3*F0*Fr^2*Xcalc^3*Xcarb^2*b*f1*r0 + 3*F0*Fr^2*Xcalc^2*Xcarb^3*b*f2*r0 - 3*F0*Fr^2*Xcalc^3*Xcarb^2*b*f2*r0 - F0*Fr^2*Xcalc*Xcarb^4*a1*f1^2*f2*r0 + F0*Fr^2*Xcalc^4*Xcarb*a1*f1^2*f2*r0 - F0*Fr^2*Xcalc*Xcarb^4*a2*f1*f2^2*r0 + F0*Fr^2*Xcalc^4*Xcarb*a2*f1*f2^2*r0 + 3*F0*Fr^2*Xcalc^2*Xcarb^3*a1*f1^2*f2*r0 - 3*F0*Fr^2*Xcalc^3*Xcarb^2*a1*f1^2*f2*r0 + 3*F0*Fr^2*Xcalc^2*Xcarb^3*a2*f1*f2^2*r0 - 3*F0*Fr^2*Xcalc^3*Xcarb^2*a2*f1*f2^2*r0

which is a cubic polynomial in z whose roots are the possible candidates of Xave.