Solve (x-p)(x-n)=(x-m)(x-q)

$$( x - p ) ( x - n ) = ( x - m ) ( x - q )$$

Solve for m

$$\left\{\begin{matrix}m=-\frac{nx+px-qx-np}{q-x}\text{, }&x\neq q\\m\in \mathrm{R}\text{, }&x=q\text{ and }\left(p=q\text{ or }n=q\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-p by x-n.

$$x^{2}-xn-px+np=\left(x-m\right)\left(x-q\right)$$

Use the distributive property to multiply x-m by x-q.

$$x^{2}-xn-px+np=x^{2}-xq-mx+mq$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-xq-mx+mq=x^{2}-xn-px+np$$

Subtract x^{2} from both sides.

$$-xq-mx+mq=x^{2}-xn-px+np-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$-xq-mx+mq=-xn-px+np$$

Add xq to both sides.

$$-mx+mq=-xn-px+np+xq$$

Combine all terms containing m.

$$\left(-x+q\right)m=-xn-px+np+xq$$

The equation is in standard form.

$$\left(q-x\right)m=qx-nx-px+np$$

Divide both sides by -x+q.

$$\frac{\left(q-x\right)m}{q-x}=\frac{qx-nx-px+np}{q-x}$$

Dividing by -x+q undoes the multiplication by -x+q.

$$m=\frac{qx-nx-px+np}{q-x}$$

Solve for n

$$\left\{\begin{matrix}n=-\frac{mx+qx-px-mq}{p-x}\text{, }&x\neq p\\n\in \mathrm{R}\text{, }&\left(x=m\text{ and }p=m\right)\text{ or }\left(x=q\text{ and }p=q\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-p by x-n.

$$x^{2}-xn-px+np=\left(x-m\right)\left(x-q\right)$$

Use the distributive property to multiply x-m by x-q.

$$x^{2}-xn-px+np=x^{2}-xq-mx+qm$$

Subtract x^{2} from both sides.

$$-xn-px+np=x^{2}-xq-mx+qm-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$-xn-px+np=-xq-mx+qm$$

Add px to both sides.

$$-xn+np=-xq-mx+qm+px$$

Combine all terms containing n.

$$\left(-x+p\right)n=-xq-mx+qm+px$$

The equation is in standard form.

$$\left(p-x\right)n=px-mx-qx+mq$$

Divide both sides by p-x.

$$\frac{\left(p-x\right)n}{p-x}=\frac{px-mx-qx+mq}{p-x}$$

Dividing by p-x undoes the multiplication by p-x.

$$n=\frac{px-mx-qx+mq}{p-x}$$