Solve pq-pr/r^2-1divq^2-r^2/r^2+r

$$\frac { p q - p r } { r ^ { 2 } - 1 } \div \frac { q ^ { 2 } - r ^ { 2 } } { r ^ { 2 } + r }$$

Evaluate

$$\frac{pr}{\left(r-1\right)\left(q+r\right)}$$

$$r\neq 0\text{ and }|r|\neq 1\text{ and }|q|\neq |r|$$

Short Solution Steps

Divide \frac{pq-pr}{r^{2}-1} by \frac{q^{2}-r^{2}}{r^{2}+r} by multiplying \frac{pq-pr}{r^{2}-1} by the reciprocal of \frac{q^{2}-r^{2}}{r^{2}+r}.

$$\frac{\left(pq-pr\right)\left(r^{2}+r\right)}{\left(r^{2}-1\right)\left(q^{2}-r^{2}\right)}$$

Factor the expressions that are not already factored.

$$\frac{pr\left(r+1\right)\left(q-r\right)}{\left(r-1\right)\left(r+1\right)\left(q+r\right)\left(q-r\right)}$$

Cancel out \left(r+1\right)\left(q-r\right) in both numerator and denominator.

$$\frac{pr}{\left(r-1\right)\left(q+r\right)}$$

Expand the expression.

$$\frac{pr}{qr-q+r^{2}-r}$$

Expand

$$\frac{pr}{\left(r-1\right)\left(q+r\right)}$$

$$r\neq 0\text{ and }|r|\neq 1\text{ and }|q|\neq |r|$$

Short Solution Steps

Divide \frac{pq-pr}{r^{2}-1} by \frac{q^{2}-r^{2}}{r^{2}+r} by multiplying \frac{pq-pr}{r^{2}-1} by the reciprocal of \frac{q^{2}-r^{2}}{r^{2}+r}.

$$\frac{\left(pq-pr\right)\left(r^{2}+r\right)}{\left(r^{2}-1\right)\left(q^{2}-r^{2}\right)}$$

Factor the expressions that are not already factored.

$$\frac{pr\left(r+1\right)\left(q-r\right)}{\left(r-1\right)\left(r+1\right)\left(q+r\right)\left(q-r\right)}$$

Cancel out \left(r+1\right)\left(q-r\right) in both numerator and denominator.

$$\frac{pr}{\left(r-1\right)\left(q+r\right)}$$

Expand the expression.

$$\frac{pr}{qr-q+r^{2}-r}$$