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

$$y = x ^ { 3 } + a x ^ { 2 } + b x + c$$

Solve for a

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

Steps for Solving Linear Equation

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

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

Subtract x^{3} from both sides.

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

Subtract bx from both sides.

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

Subtract c from both sides.

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

The equation is in standard form.

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

Divide both sides by x^{2}.

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

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

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

Solve for b

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

Steps for Solving Linear Equation

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

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

Subtract x^{3} from both sides.

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

Subtract ax^{2} from both sides.

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

Subtract c from both sides.

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

Reorder the terms.

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

The equation is in standard form.

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

Divide both sides by x.

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

Dividing by x undoes the multiplication by x.

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