Solve 9/3h^2-3k^2+h/h^2-hk

$$\frac { 9 } { 3 h ^ { 2 } - 3 k ^ { 2 } } + \frac { h } { h ^ { 2 } - h k }$$

Evaluate

$$\frac{h+k+3}{h^{2}-k^{2}}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{h}{h^{2}-hk}.

$$\frac{9}{3h^{2}-3k^{2}}+\frac{h}{h\left(h-k\right)}$$

Cancel out h in both numerator and denominator.

$$\frac{9}{3h^{2}-3k^{2}}+\frac{1}{h-k}$$

Factor 3h^{2}-3k^{2}.

$$\frac{9}{3\left(h+k\right)\left(h-k\right)}+\frac{1}{h-k}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(h+k\right)\left(h-k\right) and h-k is 3\left(h+k\right)\left(h-k\right). Multiply \frac{1}{h-k} times \frac{3\left(h+k\right)}{3\left(h+k\right)}.

$$\frac{9}{3\left(h+k\right)\left(h-k\right)}+\frac{3\left(h+k\right)}{3\left(h+k\right)\left(h-k\right)}$$

Since \frac{9}{3\left(h+k\right)\left(h-k\right)} and \frac{3\left(h+k\right)}{3\left(h+k\right)\left(h-k\right)} have the same denominator, add them by adding their numerators.

$$\frac{9+3\left(h+k\right)}{3\left(h+k\right)\left(h-k\right)}$$

Do the multiplications in 9+3\left(h+k\right).

$$\frac{9+3h+3k}{3\left(h+k\right)\left(h-k\right)}$$

Factor the expressions that are not already factored in \frac{9+3h+3k}{3\left(h+k\right)\left(h-k\right)}.

$$\frac{3\left(h+k+3\right)}{3\left(h+k\right)\left(h-k\right)}$$

Cancel out 3 in both numerator and denominator.

$$\frac{h+k+3}{\left(h+k\right)\left(h-k\right)}$$

Expand \left(h+k\right)\left(h-k\right).

$$\frac{h+k+3}{h^{2}-k^{2}}$$

Differentiate w.r.t. h

$$\frac{-h^{2}-2hk-6h-k^{2}}{\left(h^{2}-k^{2}\right)^{2}}$$