Solve x^3+px^2+q=0

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

Solve for p (complex solution)

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

Steps for Solving Linear Equation

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

$$px^{2}+q=-x^{3}$$

Subtract q from both sides.

$$px^{2}=-x^{3}-q$$

The equation is in standard form.

$$x^{2}p=-x^{3}-q$$

Divide both sides by x^{2}.

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

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

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

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

$$p=-x-\frac{q}{x^{2}}$$

Solve for p

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

Steps for Solving Linear Equation

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

$$px^{2}+q=-x^{3}$$

Subtract q from both sides.

$$px^{2}=-x^{3}-q$$

The equation is in standard form.

$$x^{2}p=-x^{3}-q$$

Divide both sides by x^{2}.

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

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

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

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

$$p=-x-\frac{q}{x^{2}}$$

Solve for q

$$q=-\left(x+p\right)x^{2}$$