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.

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

Add q to both sides.

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

The equation is in standard form.

$$\left(-x\right)p=q-x^{3}$$

Divide both sides by -x.

$$\frac{\left(-x\right)p}{-x}=\frac{q-x^{3}}{-x}$$

Dividing by -x undoes the multiplication by -x.

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

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

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

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.

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

Add q to both sides.

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

The equation is in standard form.

$$\left(-x\right)p=q-x^{3}$$

Divide both sides by -x.

$$\frac{\left(-x\right)p}{-x}=\frac{q-x^{3}}{-x}$$

Dividing by -x undoes the multiplication by -x.

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

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

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

Solve for q

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

Steps for Solving Linear Equation

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

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

Add px to both sides.

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

The equation is in standard form.

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

Divide both sides by -1.

$$\frac{-q}{-1}=\frac{x\left(p-x^{2}\right)}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$q=\frac{x\left(p-x^{2}\right)}{-1}$$

Divide x\left(-x^{2}+p\right) by -1.

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