Solve b=c*a*d/a*e+F/2(F/a*e)^2

$$b = \frac { c \cdot a \cdot d } { a \cdot e } + \frac { F } { 2 } ( \frac { F } { a \cdot e } ) ^ { 2 }$$

Solve for F (complex solution)

$$F=-a^{\frac{2}{3}}\sqrt[3]{2ecd-2e^{2}b}$$

$$F=e^{\frac{1+\pi i}{3}}a^{\frac{2}{3}}\sqrt[3]{2cd-2eb}$$

$$F=-ie^{\frac{\pi i}{6}+\frac{1}{3}}a^{\frac{2}{3}}\sqrt[3]{2cd-2eb}\text{, }a\neq 0$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{\sqrt{2}\left(e\left(eb-cd\right)\right)^{-\frac{1}{2}}F^{\frac{3}{2}}}{2}\text{; }a=\frac{\sqrt{2}\left(e\left(eb-cd\right)\right)^{-\frac{1}{2}}F^{\frac{3}{2}}}{2}\text{, }&F\neq 0\text{ and }b\neq \frac{cd}{e}\\a\neq 0\text{, }&F=0\text{ and }b=\frac{cd}{e}\end{matrix}\right.$$

Solve for F

$$F=-a^{\frac{2}{3}}\sqrt[3]{2e\left(cd-eb\right)}$$

$$a\neq 0$$