Solve x=3x^2+mx+n

$$x = 3 x ^ { 2 } + m x + n$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=-3x-\frac{n}{x}+1\text{, }&x\neq 0\\m\in \mathrm{C}\text{, }&n=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$3x^{2}+mx+n=x$$

Subtract 3x^{2} from both sides.

$$mx+n=x-3x^{2}$$

Subtract n from both sides.

$$mx=x-3x^{2}-n$$

The equation is in standard form.

$$xm=-3x^{2}+x-n$$

Divide both sides by x.

$$\frac{xm}{x}=\frac{-3x^{2}+x-n}{x}$$

Dividing by x undoes the multiplication by x.

$$m=\frac{-3x^{2}+x-n}{x}$$

Divide x-3x^{2}-n by x.

$$m=-3x-\frac{n}{x}+1$$

Solve for m

$$\left\{\begin{matrix}m=-3x-\frac{n}{x}+1\text{, }&x\neq 0\\m\in \mathrm{R}\text{, }&n=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$3x^{2}+mx+n=x$$

Subtract 3x^{2} from both sides.

$$mx+n=x-3x^{2}$$

Subtract n from both sides.

$$mx=x-3x^{2}-n$$

The equation is in standard form.

$$xm=-3x^{2}+x-n$$

Divide both sides by x.

$$\frac{xm}{x}=\frac{-3x^{2}+x-n}{x}$$

Dividing by x undoes the multiplication by x.

$$m=\frac{-3x^{2}+x-n}{x}$$

Divide x-3x^{2}-n by x.

$$m=-3x-\frac{n}{x}+1$$

Solve for n

$$n=-x\left(3x+m-1\right)$$

Solution Steps

Swap sides so that all variable terms are on the left hand side.

$$3x^{2}+mx+n=x$$

Subtract 3x^{2} from both sides.

$$mx+n=x-3x^{2}$$

Subtract mx from both sides.

$$n=x-3x^{2}-mx$$