Solve x^3+px+q=0

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

Solve for p (complex solution)

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

Steps for Solving Linear Equation

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

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

Subtract q from both sides.

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

The equation is in standard form.

$$xp=-x^{3}-q$$

Divide both sides by x.

$$\frac{xp}{x}=\frac{-x^{3}-q}{x}$$

Dividing by x undoes the multiplication by x.

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

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

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

Solve for p

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

Steps for Solving Linear Equation

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

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

Subtract q from both sides.

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

The equation is in standard form.

$$xp=-x^{3}-q$$

Divide both sides by x.

$$\frac{xp}{x}=\frac{-x^{3}-q}{x}$$

Dividing by x undoes the multiplication by x.

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

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

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

Solve for q

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

Solution Steps

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

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

Subtract px from both sides.

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