Solve x^3+3px^2+3qx+r=0

$$x ^ { 3 } + 3 p x ^ { 2 } + 3 q x + r = 0$$

Solve for p

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

Steps for Solving Linear Equation

Subtract x^{3} from both sides. Anything subtracted from zero gives its negation.

$$3px^{2}+3qx+r=-x^{3}$$

Subtract 3qx from both sides.

$$3px^{2}+r=-x^{3}-3qx$$

Subtract r from both sides.

$$3px^{2}=-x^{3}-3qx-r$$

The equation is in standard form.

$$3x^{2}p=-x^{3}-3qx-r$$

Divide both sides by 3x^{2}.

$$\frac{3x^{2}p}{3x^{2}}=\frac{-x^{3}-3qx-r}{3x^{2}}$$

Dividing by 3x^{2} undoes the multiplication by 3x^{2}.

$$p=\frac{-x^{3}-3qx-r}{3x^{2}}$$

Divide -x^{3}-3qx-r by 3x^{2}.

$$p=-\frac{qx+\frac{r}{3}}{x^{2}}-\frac{x}{3}$$

Solve for q

$$\left\{\begin{matrix}q=-px-\frac{x^{2}}{3}-\frac{r}{3x}\text{, }&x\neq 0\\q\in \mathrm{R}\text{, }&x=0\text{ and }r=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract x^{3} from both sides. Anything subtracted from zero gives its negation.

$$3px^{2}+3qx+r=-x^{3}$$

Subtract 3px^{2} from both sides.

$$3qx+r=-x^{3}-3px^{2}$$

Subtract r from both sides.

$$3qx=-x^{3}-3px^{2}-r$$

The equation is in standard form.

$$3xq=-x^{3}-3px^{2}-r$$

Divide both sides by 3x.

$$\frac{3xq}{3x}=\frac{-x^{3}-3px^{2}-r}{3x}$$

Dividing by 3x undoes the multiplication by 3x.

$$q=\frac{-x^{3}-3px^{2}-r}{3x}$$

Divide -x^{3}-3px^{2}-r by 3x.

$$q=-px-\frac{x^{2}}{3}-\frac{r}{3x}$$