Solve Pa(p-q)+d(p+q)(p-q)-d(p+q)=0

$$P a ( p - q ) + d ( p + q ) ( p - q ) - d ( p + q ) = 0$$

Solve for P (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply Pa by p-q.

$$Pap-Paq+d\left(p+q\right)\left(p-q\right)-d\left(p+q\right)=0$$

Use the distributive property to multiply d by p+q.

$$Pap-Paq+\left(dp+dq\right)\left(p-q\right)-d\left(p+q\right)=0$$

Use the distributive property to multiply dp+dq by p-q and combine like terms.

$$Pap-Paq+dp^{2}-dq^{2}-d\left(p+q\right)=0$$

Use the distributive property to multiply d by p+q.

$$Pap-Paq+dp^{2}-dq^{2}-\left(dp+dq\right)=0$$

To find the opposite of dp+dq, find the opposite of each term.

$$Pap-Paq+dp^{2}-dq^{2}-dp-dq=0$$

Subtract dp^{2} from both sides. Anything subtracted from zero gives its negation.

$$Pap-Paq-dq^{2}-dp-dq=-dp^{2}$$

Add dq^{2} to both sides.

$$Pap-Paq-dp-dq=-dp^{2}+dq^{2}$$

Add dp to both sides.

$$Pap-Paq-dq=-dp^{2}+dq^{2}+dp$$

Add dq to both sides.

$$Pap-Paq=-dp^{2}+dq^{2}+dp+dq$$

Reorder the terms.

$$Pap-Paq=-dp^{2}+dp+dq^{2}+dq$$

Combine all terms containing P.

$$\left(ap-aq\right)P=-dp^{2}+dp+dq^{2}+dq$$

The equation is in standard form.

$$\left(ap-aq\right)P=dq+dq^{2}+dp-dp^{2}$$

Divide both sides by ap-aq.

$$\frac{\left(ap-aq\right)P}{ap-aq}=-\frac{d\left(p-q-1\right)\left(p+q\right)}{ap-aq}$$

Dividing by ap-aq undoes the multiplication by ap-aq.

$$P=-\frac{d\left(p-q-1\right)\left(p+q\right)}{ap-aq}$$

Divide -d\left(-1+p-q\right)\left(p+q\right) by ap-aq.

$$P=-\frac{d\left(p-q-1\right)\left(p+q\right)}{a\left(p-q\right)}$$

Solve for a (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply Pa by p-q.

$$Pap-Paq+d\left(p+q\right)\left(p-q\right)-d\left(p+q\right)=0$$

Use the distributive property to multiply d by p+q.

$$Pap-Paq+\left(dp+dq\right)\left(p-q\right)-d\left(p+q\right)=0$$

Use the distributive property to multiply dp+dq by p-q and combine like terms.

$$Pap-Paq+dp^{2}-dq^{2}-d\left(p+q\right)=0$$

Use the distributive property to multiply d by p+q.

$$Pap-Paq+dp^{2}-dq^{2}-\left(dp+dq\right)=0$$

To find the opposite of dp+dq, find the opposite of each term.

$$Pap-Paq+dp^{2}-dq^{2}-dp-dq=0$$

Subtract dp^{2} from both sides. Anything subtracted from zero gives its negation.

$$Pap-Paq-dq^{2}-dp-dq=-dp^{2}$$

Add dq^{2} to both sides.

$$Pap-Paq-dp-dq=-dp^{2}+dq^{2}$$

Add dp to both sides.

$$Pap-Paq-dq=-dp^{2}+dq^{2}+dp$$

Add dq to both sides.

$$Pap-Paq=-dp^{2}+dq^{2}+dp+dq$$

Reorder the terms.

$$Pap-Paq=-dp^{2}+dp+dq^{2}+dq$$

Combine all terms containing a.

$$\left(Pp-Pq\right)a=-dp^{2}+dp+dq^{2}+dq$$

The equation is in standard form.

$$\left(Pp-Pq\right)a=dq+dq^{2}+dp-dp^{2}$$

Divide both sides by Pp-Pq.

$$\frac{\left(Pp-Pq\right)a}{Pp-Pq}=-\frac{d\left(p-q-1\right)\left(p+q\right)}{Pp-Pq}$$

Dividing by Pp-Pq undoes the multiplication by Pp-Pq.

$$a=-\frac{d\left(p-q-1\right)\left(p+q\right)}{Pp-Pq}$$

Divide -d\left(-1+p-q\right)\left(p+q\right) by Pp-Pq.

$$a=-\frac{d\left(p-q-1\right)\left(p+q\right)}{P\left(p-q\right)}$$

Solve for P

$$\left\{\begin{matrix}P=-\frac{d\left(p-q-1\right)\left(p+q\right)}{a\left(p-q\right)}\text{, }&p\neq q\text{ and }a\neq 0\\P\in \mathrm{R}\text{, }&\left(d=0\text{ and }p=q\right)\text{ or }\left(p=0\text{ and }q=0\right)\text{ or }\left(a=0\text{ and }p=q+1\right)\text{ or }\left(p=-q\text{ and }a=0\text{ and }q\neq 0\right)\text{ or }\left(d=0\text{ and }a=0\text{ and }p\neq q\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply Pa by p-q.

$$Pap-Paq+d\left(p+q\right)\left(p-q\right)-d\left(p+q\right)=0$$

Use the distributive property to multiply d by p+q.

$$Pap-Paq+\left(dp+dq\right)\left(p-q\right)-d\left(p+q\right)=0$$

Use the distributive property to multiply dp+dq by p-q and combine like terms.

$$Pap-Paq+dp^{2}-dq^{2}-d\left(p+q\right)=0$$

Use the distributive property to multiply d by p+q.

$$Pap-Paq+dp^{2}-dq^{2}-\left(dp+dq\right)=0$$

To find the opposite of dp+dq, find the opposite of each term.

$$Pap-Paq+dp^{2}-dq^{2}-dp-dq=0$$

Subtract dp^{2} from both sides. Anything subtracted from zero gives its negation.

$$Pap-Paq-dq^{2}-dp-dq=-dp^{2}$$

Add dq^{2} to both sides.

$$Pap-Paq-dp-dq=-dp^{2}+dq^{2}$$

Add dp to both sides.

$$Pap-Paq-dq=-dp^{2}+dq^{2}+dp$$

Add dq to both sides.

$$Pap-Paq=-dp^{2}+dq^{2}+dp+dq$$

Reorder the terms.

$$Pap-Paq=-dp^{2}+dp+dq^{2}+dq$$

Combine all terms containing P.

$$\left(ap-aq\right)P=-dp^{2}+dp+dq^{2}+dq$$

The equation is in standard form.

$$\left(ap-aq\right)P=dq+dq^{2}+dp-dp^{2}$$

Divide both sides by ap-aq.

$$\frac{\left(ap-aq\right)P}{ap-aq}=-\frac{d\left(p-q-1\right)\left(p+q\right)}{ap-aq}$$

Dividing by ap-aq undoes the multiplication by ap-aq.

$$P=-\frac{d\left(p-q-1\right)\left(p+q\right)}{ap-aq}$$

Divide -d\left(-1+p-q\right)\left(p+q\right) by ap-aq.

$$P=-\frac{d\left(p-q-1\right)\left(p+q\right)}{a\left(p-q\right)}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{d\left(p-q-1\right)\left(p+q\right)}{P\left(p-q\right)}\text{, }&p\neq q\text{ and }P\neq 0\\a\in \mathrm{R}\text{, }&\left(d=0\text{ and }p=q\right)\text{ or }\left(p=0\text{ and }q=0\right)\text{ or }\left(P=0\text{ and }p=q+1\right)\text{ or }\left(p=-q\text{ and }P=0\text{ and }q\neq 0\right)\text{ or }\left(d=0\text{ and }P=0\text{ and }p\neq q\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply Pa by p-q.

$$Pap-Paq+d\left(p+q\right)\left(p-q\right)-d\left(p+q\right)=0$$

Use the distributive property to multiply d by p+q.

$$Pap-Paq+\left(dp+dq\right)\left(p-q\right)-d\left(p+q\right)=0$$

Use the distributive property to multiply dp+dq by p-q and combine like terms.

$$Pap-Paq+dp^{2}-dq^{2}-d\left(p+q\right)=0$$

Use the distributive property to multiply d by p+q.

$$Pap-Paq+dp^{2}-dq^{2}-\left(dp+dq\right)=0$$

To find the opposite of dp+dq, find the opposite of each term.

$$Pap-Paq+dp^{2}-dq^{2}-dp-dq=0$$

Subtract dp^{2} from both sides. Anything subtracted from zero gives its negation.

$$Pap-Paq-dq^{2}-dp-dq=-dp^{2}$$

Add dq^{2} to both sides.

$$Pap-Paq-dp-dq=-dp^{2}+dq^{2}$$

Add dp to both sides.

$$Pap-Paq-dq=-dp^{2}+dq^{2}+dp$$

Add dq to both sides.

$$Pap-Paq=-dp^{2}+dq^{2}+dp+dq$$

Reorder the terms.

$$Pap-Paq=-dp^{2}+dp+dq^{2}+dq$$

Combine all terms containing a.

$$\left(Pp-Pq\right)a=-dp^{2}+dp+dq^{2}+dq$$

The equation is in standard form.

$$\left(Pp-Pq\right)a=dq+dq^{2}+dp-dp^{2}$$

Divide both sides by Pp-Pq.

$$\frac{\left(Pp-Pq\right)a}{Pp-Pq}=-\frac{d\left(p-q-1\right)\left(p+q\right)}{Pp-Pq}$$

Dividing by Pp-Pq undoes the multiplication by Pp-Pq.

$$a=-\frac{d\left(p-q-1\right)\left(p+q\right)}{Pp-Pq}$$

Divide -d\left(-1+p-q\right)\left(p+q\right) by Pp-Pq.

$$a=-\frac{d\left(p-q-1\right)\left(p+q\right)}{P\left(p-q\right)}$$