Simplify \(\frac{10}{2}\) to \(5\).
\[\begin{aligned}&Fx=5{x}^{3}+{x}^{2}-3x\\&gx=5{x}^{3-3x}+{x}^{2}\end{aligned}\]
Break down the problem into these 2 equations.
\[Fx=5{x}^{3}+{x}^{2}-3x\]
\[Fx=gx\]
Solve the 1st equation: \(Fx=5{x}^{3}+{x}^{2}-3x\).
Move all terms to one side.
\[Fx-5{x}^{3}-{x}^{2}+3x=0\]
Factor out the common term \(x\).
\[x(F-5{x}^{2}-x+3)=0\]
Solve for \(x\).
\[x=0\]
Use the Quadratic Formula.
In general, given \(a{x}^{2}+bx+c=0\), there exists two solutions where:
\[x=\frac{-b+\sqrt{{b}^{2}-4ac}}{2a},\frac{-b-\sqrt{{b}^{2}-4ac}}{2a}\]
In this case, \(a=-5\), \(b=-1\) and \(c=F+3\).
\[{x}^{}=\frac{1+\sqrt{{(-1)}^{2}-4\times -5(F+3)}}{2\times -5},\frac{1-\sqrt{{(-1)}^{2}-4\times -5(F+3)}}{2\times -5}\]
Simplify.
\[x=\frac{1+\sqrt{20F+61}}{-10},\frac{1-\sqrt{20F+61}}{-10}\]
\[x=\frac{1+\sqrt{20F+61}}{-10},\frac{1-\sqrt{20F+61}}{-10}\]
Collect all solutions from the previous steps.
\[x=0,\frac{1+\sqrt{20F+61}}{-10},\frac{1-\sqrt{20F+61}}{-10}\]
Simplify solutions.
\[x=0,-\frac{1+\sqrt{20F+61}}{10},-\frac{1-\sqrt{20F+61}}{10}\]
\[x=0,-\frac{1+\sqrt{20F+61}}{10},-\frac{1-\sqrt{20F+61}}{10}\]
Solve the 2nd equation: \(Fx=gx\).
Move all terms to one side.
\[Fx-gx=0\]
Factor out the common term \(x\).
\[x(F-g)=0\]
Divide both sides by \(F-g\).
\[x=0\]
\[x=0\]
Collect all solutions.
\[x=0,-\frac{1+\sqrt{20F+61}}{10},-\frac{1-\sqrt{20F+61}}{10},0\]
Therefore,
\(x=0,-\frac{1+\sqrt{20F+61}}{10},-\frac{1-\sqrt{20F+61}}{10}\)
x=0,-(1+sqrt(20*F+61))/10,-(1-sqrt(20*F+61))/10
