Solve (8b/b+7-15b/b^2+14b+49):8b+41/b^2-49+7b-4/6+7=

$$( \frac { 8 b } { b + 7 } - \frac { 15 b } { b ^ { 2 } + 14 b + 49 } ) : \frac { 8 b + 41 } { b ^ { 2 } - 49 } + \frac { 7 b - 4 } { 6 + 7 } =$$

Evaluate

$$\frac{2\left(5b-14\right)\left(2b+1\right)}{13\left(b+7\right)}$$

Short Solution Steps

Factor b^{2}+14b+49.

$$\frac{\frac{8b}{b+7}-\frac{15b}{\left(b+7\right)^{2}}}{\frac{8b+41}{b^{2}-49}}+\frac{7b-4}{6+7}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b+7 and \left(b+7\right)^{2} is \left(b+7\right)^{2}. Multiply \frac{8b}{b+7} times \frac{b+7}{b+7}.

$$\frac{\frac{8b\left(b+7\right)}{\left(b+7\right)^{2}}-\frac{15b}{\left(b+7\right)^{2}}}{\frac{8b+41}{b^{2}-49}}+\frac{7b-4}{6+7}$$

Since \frac{8b\left(b+7\right)}{\left(b+7\right)^{2}} and \frac{15b}{\left(b+7\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{8b\left(b+7\right)-15b}{\left(b+7\right)^{2}}}{\frac{8b+41}{b^{2}-49}}+\frac{7b-4}{6+7}$$

Do the multiplications in 8b\left(b+7\right)-15b.

$$\frac{\frac{8b^{2}+56b-15b}{\left(b+7\right)^{2}}}{\frac{8b+41}{b^{2}-49}}+\frac{7b-4}{6+7}$$

Combine like terms in 8b^{2}+56b-15b.

$$\frac{\frac{8b^{2}+41b}{\left(b+7\right)^{2}}}{\frac{8b+41}{b^{2}-49}}+\frac{7b-4}{6+7}$$

Divide \frac{8b^{2}+41b}{\left(b+7\right)^{2}} by \frac{8b+41}{b^{2}-49} by multiplying \frac{8b^{2}+41b}{\left(b+7\right)^{2}} by the reciprocal of \frac{8b+41}{b^{2}-49}.

$$\frac{\left(8b^{2}+41b\right)\left(b^{2}-49\right)}{\left(b+7\right)^{2}\left(8b+41\right)}+\frac{7b-4}{6+7}$$

Factor the expressions that are not already factored in \frac{\left(8b^{2}+41b\right)\left(b^{2}-49\right)}{\left(b+7\right)^{2}\left(8b+41\right)}.

$$\frac{b\left(b-7\right)\left(b+7\right)\left(8b+41\right)}{\left(8b+41\right)\left(b+7\right)^{2}}+\frac{7b-4}{6+7}$$

Cancel out \left(b+7\right)\left(8b+41\right) in both numerator and denominator.

$$\frac{b\left(b-7\right)}{b+7}+\frac{7b-4}{6+7}$$

Add 6 and 7 to get 13.

$$\frac{b\left(b-7\right)}{b+7}+\frac{7b-4}{13}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b+7 and 13 is 13\left(b+7\right). Multiply \frac{b\left(b-7\right)}{b+7} times \frac{13}{13}. Multiply \frac{7b-4}{13} times \frac{b+7}{b+7}.

$$\frac{13b\left(b-7\right)}{13\left(b+7\right)}+\frac{\left(7b-4\right)\left(b+7\right)}{13\left(b+7\right)}$$

Since \frac{13b\left(b-7\right)}{13\left(b+7\right)} and \frac{\left(7b-4\right)\left(b+7\right)}{13\left(b+7\right)} have the same denominator, add them by adding their numerators.

$$\frac{13b\left(b-7\right)+\left(7b-4\right)\left(b+7\right)}{13\left(b+7\right)}$$

Do the multiplications in 13b\left(b-7\right)+\left(7b-4\right)\left(b+7\right).

$$\frac{13b^{2}-91b+7b^{2}+49b-4b-28}{13\left(b+7\right)}$$

Combine like terms in 13b^{2}-91b+7b^{2}+49b-4b-28.

$$\frac{20b^{2}-46b-28}{13\left(b+7\right)}$$

Expand 13\left(b+7\right).

$$\frac{20b^{2}-46b-28}{13b+91}$$

Expand

$$\frac{2\left(10b^{2}-23b-14\right)}{13\left(b+7\right)}$$

Short Solution Steps

Factor b^{2}+14b+49.

$$\frac{\frac{8b}{b+7}-\frac{15b}{\left(b+7\right)^{2}}}{\frac{8b+41}{b^{2}-49}}+\frac{7b-4}{6+7}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b+7 and \left(b+7\right)^{2} is \left(b+7\right)^{2}. Multiply \frac{8b}{b+7} times \frac{b+7}{b+7}.

$$\frac{\frac{8b\left(b+7\right)}{\left(b+7\right)^{2}}-\frac{15b}{\left(b+7\right)^{2}}}{\frac{8b+41}{b^{2}-49}}+\frac{7b-4}{6+7}$$

Since \frac{8b\left(b+7\right)}{\left(b+7\right)^{2}} and \frac{15b}{\left(b+7\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{8b\left(b+7\right)-15b}{\left(b+7\right)^{2}}}{\frac{8b+41}{b^{2}-49}}+\frac{7b-4}{6+7}$$

Do the multiplications in 8b\left(b+7\right)-15b.

$$\frac{\frac{8b^{2}+56b-15b}{\left(b+7\right)^{2}}}{\frac{8b+41}{b^{2}-49}}+\frac{7b-4}{6+7}$$

Combine like terms in 8b^{2}+56b-15b.

$$\frac{\frac{8b^{2}+41b}{\left(b+7\right)^{2}}}{\frac{8b+41}{b^{2}-49}}+\frac{7b-4}{6+7}$$

Divide \frac{8b^{2}+41b}{\left(b+7\right)^{2}} by \frac{8b+41}{b^{2}-49} by multiplying \frac{8b^{2}+41b}{\left(b+7\right)^{2}} by the reciprocal of \frac{8b+41}{b^{2}-49}.

$$\frac{\left(8b^{2}+41b\right)\left(b^{2}-49\right)}{\left(b+7\right)^{2}\left(8b+41\right)}+\frac{7b-4}{6+7}$$

Factor the expressions that are not already factored in \frac{\left(8b^{2}+41b\right)\left(b^{2}-49\right)}{\left(b+7\right)^{2}\left(8b+41\right)}.

$$\frac{b\left(b-7\right)\left(b+7\right)\left(8b+41\right)}{\left(8b+41\right)\left(b+7\right)^{2}}+\frac{7b-4}{6+7}$$

Cancel out \left(b+7\right)\left(8b+41\right) in both numerator and denominator.

$$\frac{b\left(b-7\right)}{b+7}+\frac{7b-4}{6+7}$$

Add 6 and 7 to get 13.

$$\frac{b\left(b-7\right)}{b+7}+\frac{7b-4}{13}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b+7 and 13 is 13\left(b+7\right). Multiply \frac{b\left(b-7\right)}{b+7} times \frac{13}{13}. Multiply \frac{7b-4}{13} times \frac{b+7}{b+7}.

$$\frac{13b\left(b-7\right)}{13\left(b+7\right)}+\frac{\left(7b-4\right)\left(b+7\right)}{13\left(b+7\right)}$$

Since \frac{13b\left(b-7\right)}{13\left(b+7\right)} and \frac{\left(7b-4\right)\left(b+7\right)}{13\left(b+7\right)} have the same denominator, add them by adding their numerators.

$$\frac{13b\left(b-7\right)+\left(7b-4\right)\left(b+7\right)}{13\left(b+7\right)}$$

Do the multiplications in 13b\left(b-7\right)+\left(7b-4\right)\left(b+7\right).

$$\frac{13b^{2}-91b+7b^{2}+49b-4b-28}{13\left(b+7\right)}$$

Combine like terms in 13b^{2}-91b+7b^{2}+49b-4b-28.

$$\frac{20b^{2}-46b-28}{13\left(b+7\right)}$$

Expand 13\left(b+7\right).

$$\frac{20b^{2}-46b-28}{13b+91}$$