Solve px^2+(q+1)x+1-q^2=0

$$p x ^ { 2 } + ( q + 1 ) x + 1 - q ^ { 2 } = 0$$

Solve for p (complex solution)

$$\left\{\begin{matrix}p=\frac{\left(q+1\right)\left(-x+q-1\right)}{x^{2}}\text{, }&x\neq 0\\p\in \mathrm{C}\text{, }&\left(q=1\text{ or }q=-1\right)\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply q+1 by x.

$$px^{2}+qx+x+1-q^{2}=0$$

Subtract qx from both sides. Anything subtracted from zero gives its negation.

$$px^{2}+x+1-q^{2}=-qx$$

Subtract x from both sides.

$$px^{2}+1-q^{2}=-qx-x$$

Subtract 1 from both sides.

$$px^{2}-q^{2}=-qx-x-1$$

Add q^{2} to both sides.

$$px^{2}=-qx-x-1+q^{2}$$

Reorder the terms.

$$px^{2}=-qx-x+q^{2}-1$$

The equation is in standard form.

$$x^{2}p=-qx-x+q^{2}-1$$

Divide both sides by x^{2}.

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

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

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

Solve for p

$$\left\{\begin{matrix}p=\frac{\left(q+1\right)\left(-x+q-1\right)}{x^{2}}\text{, }&x\neq 0\\p\in \mathrm{R}\text{, }&x=0\text{ and }|q|=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply q+1 by x.

$$px^{2}+qx+x+1-q^{2}=0$$

Subtract qx from both sides. Anything subtracted from zero gives its negation.

$$px^{2}+x+1-q^{2}=-qx$$

Subtract x from both sides.

$$px^{2}+1-q^{2}=-qx-x$$

Subtract 1 from both sides.

$$px^{2}-q^{2}=-qx-x-1$$

Add q^{2} to both sides.

$$px^{2}=-qx-x-1+q^{2}$$

Reorder the terms.

$$px^{2}=-qx-x+q^{2}-1$$

The equation is in standard form.

$$x^{2}p=-qx-x+q^{2}-1$$

Divide both sides by x^{2}.

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

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

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

Solve for q (complex solution)

$$q=\frac{-\sqrt{4px^{2}+x^{2}+4x+4}+x}{2}$$

$$q=\frac{\sqrt{4px^{2}+x^{2}+4x+4}+x}{2}$$

Solve for q

$$q=\frac{-\sqrt{4px^{2}+x^{2}+4x+4}+x}{2}$$

$$q=\frac{\sqrt{4px^{2}+x^{2}+4x+4}+x}{2}\text{, }p\geq -\frac{\left(x+2\right)^{2}}{4x^{2}}\text{ or }x=0$$