Solve x^3+ax^2+bx+c=2

$${ x }^{ 3 } +a { x }^{ 2 } +bx+c = 2$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{x^{3}+bx+c-2}{x^{2}}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&x=0\text{ and }c=2\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract x^{3} from both sides.

$$ax^{2}+bx+c=2-x^{3}$$

Subtract bx from both sides.

$$ax^{2}+c=2-x^{3}-bx$$

Subtract c from both sides.

$$ax^{2}=2-x^{3}-bx-c$$

The equation is in standard form.

$$x^{2}a=2-c-bx-x^{3}$$

Divide both sides by x^{2}.

$$\frac{x^{2}a}{x^{2}}=\frac{2-c-bx-x^{3}}{x^{2}}$$

Dividing by x^{2} undoes the multiplication by x^{2}.

$$a=\frac{2-c-bx-x^{3}}{x^{2}}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{x^{3}+ax^{2}+c-2}{x}\text{, }&x\neq 0\\b\in \mathrm{R}\text{, }&x=0\text{ and }c=2\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract x^{3} from both sides.

$$ax^{2}+bx+c=2-x^{3}$$

Subtract ax^{2} from both sides.

$$bx+c=2-x^{3}-ax^{2}$$

Subtract c from both sides.

$$bx=2-x^{3}-ax^{2}-c$$

Reorder the terms.

$$bx=-x^{3}-ax^{2}-c+2$$

The equation is in standard form.

$$xb=2-c-ax^{2}-x^{3}$$

Divide both sides by x.

$$\frac{xb}{x}=\frac{2-c-ax^{2}-x^{3}}{x}$$

Dividing by x undoes the multiplication by x.

$$b=\frac{2-c-ax^{2}-x^{3}}{x}$$