Solve 15/b^2-4+2/b+2/frac{17{b^2-4}+15/b-2}

$$\frac{ \frac{ 15 }{ { b }^{ 2 } -4 } + \frac{ 2 }{ b+2 } }{ \frac{ 17 }{ { b }^{ 2 } -4 } + \frac{ 15 }{ b-2 } }$$

Evaluate

$$\frac{2b+11}{15b+47}$$

Short Solution Steps

Factor b^{2}-4.

$$\frac{\frac{15}{\left(b-2\right)\left(b+2\right)}+\frac{2}{b+2}}{\frac{17}{b^{2}-4}+\frac{15}{b-2}}$$

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

$$\frac{\frac{15}{\left(b-2\right)\left(b+2\right)}+\frac{2\left(b-2\right)}{\left(b-2\right)\left(b+2\right)}}{\frac{17}{b^{2}-4}+\frac{15}{b-2}}$$

Since \frac{15}{\left(b-2\right)\left(b+2\right)} and \frac{2\left(b-2\right)}{\left(b-2\right)\left(b+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{15+2\left(b-2\right)}{\left(b-2\right)\left(b+2\right)}}{\frac{17}{b^{2}-4}+\frac{15}{b-2}}$$

Do the multiplications in 15+2\left(b-2\right).

$$\frac{\frac{15+2b-4}{\left(b-2\right)\left(b+2\right)}}{\frac{17}{b^{2}-4}+\frac{15}{b-2}}$$

Combine like terms in 15+2b-4.

$$\frac{\frac{11+2b}{\left(b-2\right)\left(b+2\right)}}{\frac{17}{b^{2}-4}+\frac{15}{b-2}}$$

Factor b^{2}-4.

$$\frac{\frac{11+2b}{\left(b-2\right)\left(b+2\right)}}{\frac{17}{\left(b-2\right)\left(b+2\right)}+\frac{15}{b-2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(b-2\right)\left(b+2\right) and b-2 is \left(b-2\right)\left(b+2\right). Multiply \frac{15}{b-2} times \frac{b+2}{b+2}.

$$\frac{\frac{11+2b}{\left(b-2\right)\left(b+2\right)}}{\frac{17}{\left(b-2\right)\left(b+2\right)}+\frac{15\left(b+2\right)}{\left(b-2\right)\left(b+2\right)}}$$

Since \frac{17}{\left(b-2\right)\left(b+2\right)} and \frac{15\left(b+2\right)}{\left(b-2\right)\left(b+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{11+2b}{\left(b-2\right)\left(b+2\right)}}{\frac{17+15\left(b+2\right)}{\left(b-2\right)\left(b+2\right)}}$$

Do the multiplications in 17+15\left(b+2\right).

$$\frac{\frac{11+2b}{\left(b-2\right)\left(b+2\right)}}{\frac{17+15b+30}{\left(b-2\right)\left(b+2\right)}}$$

Combine like terms in 17+15b+30.

$$\frac{\frac{11+2b}{\left(b-2\right)\left(b+2\right)}}{\frac{47+15b}{\left(b-2\right)\left(b+2\right)}}$$

Divide \frac{11+2b}{\left(b-2\right)\left(b+2\right)} by \frac{47+15b}{\left(b-2\right)\left(b+2\right)} by multiplying \frac{11+2b}{\left(b-2\right)\left(b+2\right)} by the reciprocal of \frac{47+15b}{\left(b-2\right)\left(b+2\right)}.

$$\frac{\left(11+2b\right)\left(b-2\right)\left(b+2\right)}{\left(b-2\right)\left(b+2\right)\left(47+15b\right)}$$

Cancel out \left(b-2\right)\left(b+2\right) in both numerator and denominator.

$$\frac{2b+11}{15b+47}$$

Expand

$$\frac{2b+11}{15b+47}$$

Short Solution Steps

Factor b^{2}-4.

$$\frac{\frac{15}{\left(b-2\right)\left(b+2\right)}+\frac{2}{b+2}}{\frac{17}{b^{2}-4}+\frac{15}{b-2}}$$

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

$$\frac{\frac{15}{\left(b-2\right)\left(b+2\right)}+\frac{2\left(b-2\right)}{\left(b-2\right)\left(b+2\right)}}{\frac{17}{b^{2}-4}+\frac{15}{b-2}}$$

Since \frac{15}{\left(b-2\right)\left(b+2\right)} and \frac{2\left(b-2\right)}{\left(b-2\right)\left(b+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{15+2\left(b-2\right)}{\left(b-2\right)\left(b+2\right)}}{\frac{17}{b^{2}-4}+\frac{15}{b-2}}$$

Do the multiplications in 15+2\left(b-2\right).

$$\frac{\frac{15+2b-4}{\left(b-2\right)\left(b+2\right)}}{\frac{17}{b^{2}-4}+\frac{15}{b-2}}$$

Combine like terms in 15+2b-4.

$$\frac{\frac{11+2b}{\left(b-2\right)\left(b+2\right)}}{\frac{17}{b^{2}-4}+\frac{15}{b-2}}$$

Factor b^{2}-4.

$$\frac{\frac{11+2b}{\left(b-2\right)\left(b+2\right)}}{\frac{17}{\left(b-2\right)\left(b+2\right)}+\frac{15}{b-2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(b-2\right)\left(b+2\right) and b-2 is \left(b-2\right)\left(b+2\right). Multiply \frac{15}{b-2} times \frac{b+2}{b+2}.

$$\frac{\frac{11+2b}{\left(b-2\right)\left(b+2\right)}}{\frac{17}{\left(b-2\right)\left(b+2\right)}+\frac{15\left(b+2\right)}{\left(b-2\right)\left(b+2\right)}}$$

Since \frac{17}{\left(b-2\right)\left(b+2\right)} and \frac{15\left(b+2\right)}{\left(b-2\right)\left(b+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{11+2b}{\left(b-2\right)\left(b+2\right)}}{\frac{17+15\left(b+2\right)}{\left(b-2\right)\left(b+2\right)}}$$

Do the multiplications in 17+15\left(b+2\right).

$$\frac{\frac{11+2b}{\left(b-2\right)\left(b+2\right)}}{\frac{17+15b+30}{\left(b-2\right)\left(b+2\right)}}$$

Combine like terms in 17+15b+30.

$$\frac{\frac{11+2b}{\left(b-2\right)\left(b+2\right)}}{\frac{47+15b}{\left(b-2\right)\left(b+2\right)}}$$

Divide \frac{11+2b}{\left(b-2\right)\left(b+2\right)} by \frac{47+15b}{\left(b-2\right)\left(b+2\right)} by multiplying \frac{11+2b}{\left(b-2\right)\left(b+2\right)} by the reciprocal of \frac{47+15b}{\left(b-2\right)\left(b+2\right)}.

$$\frac{\left(11+2b\right)\left(b-2\right)\left(b+2\right)}{\left(b-2\right)\left(b+2\right)\left(47+15b\right)}$$

Cancel out \left(b-2\right)\left(b+2\right) in both numerator and denominator.

$$\frac{2b+11}{15b+47}$$