Solve (2/c-2+3c-21/c^2+c-6+2c/c+3)c/2c-5

$$( \frac { 2 } { c - 2 } + \frac { 3 c - 21 } { c ^ { 2 } + c - 6 } + \frac { 2 c } { c + 3 } ) \frac { c } { 2 c - 5 }$$

Evaluate

$$\frac{c}{c-2}$$

Short Solution Steps

Factor c^{2}+c-6.

$$\left(\frac{2}{c-2}+\frac{3c-21}{\left(c-2\right)\left(c+3\right)}+\frac{2c}{c+3}\right)\times \frac{c}{2c-5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of c-2 and \left(c-2\right)\left(c+3\right) is \left(c-2\right)\left(c+3\right). Multiply \frac{2}{c-2} times \frac{c+3}{c+3}.

$$\left(\frac{2\left(c+3\right)}{\left(c-2\right)\left(c+3\right)}+\frac{3c-21}{\left(c-2\right)\left(c+3\right)}+\frac{2c}{c+3}\right)\times \frac{c}{2c-5}$$

Since \frac{2\left(c+3\right)}{\left(c-2\right)\left(c+3\right)} and \frac{3c-21}{\left(c-2\right)\left(c+3\right)} have the same denominator, add them by adding their numerators.

$$\left(\frac{2\left(c+3\right)+3c-21}{\left(c-2\right)\left(c+3\right)}+\frac{2c}{c+3}\right)\times \frac{c}{2c-5}$$

Do the multiplications in 2\left(c+3\right)+3c-21.

$$\left(\frac{2c+6+3c-21}{\left(c-2\right)\left(c+3\right)}+\frac{2c}{c+3}\right)\times \frac{c}{2c-5}$$

Combine like terms in 2c+6+3c-21.

$$\left(\frac{5c-15}{\left(c-2\right)\left(c+3\right)}+\frac{2c}{c+3}\right)\times \frac{c}{2c-5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(c-2\right)\left(c+3\right) and c+3 is \left(c-2\right)\left(c+3\right). Multiply \frac{2c}{c+3} times \frac{c-2}{c-2}.

$$\left(\frac{5c-15}{\left(c-2\right)\left(c+3\right)}+\frac{2c\left(c-2\right)}{\left(c-2\right)\left(c+3\right)}\right)\times \frac{c}{2c-5}$$

Since \frac{5c-15}{\left(c-2\right)\left(c+3\right)} and \frac{2c\left(c-2\right)}{\left(c-2\right)\left(c+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{5c-15+2c\left(c-2\right)}{\left(c-2\right)\left(c+3\right)}\times \frac{c}{2c-5}$$

Do the multiplications in 5c-15+2c\left(c-2\right).

$$\frac{5c-15+2c^{2}-4c}{\left(c-2\right)\left(c+3\right)}\times \frac{c}{2c-5}$$

Combine like terms in 5c-15+2c^{2}-4c.

$$\frac{c-15+2c^{2}}{\left(c-2\right)\left(c+3\right)}\times \frac{c}{2c-5}$$

Factor the expressions that are not already factored in \frac{c-15+2c^{2}}{\left(c-2\right)\left(c+3\right)}.

$$\frac{\left(2c-5\right)\left(c+3\right)}{\left(c-2\right)\left(c+3\right)}\times \frac{c}{2c-5}$$

Cancel out c+3 in both numerator and denominator.

$$\frac{2c-5}{c-2}\times \frac{c}{2c-5}$$

Multiply \frac{2c-5}{c-2} times \frac{c}{2c-5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(2c-5\right)c}{\left(c-2\right)\left(2c-5\right)}$$

Cancel out 2c-5 in both numerator and denominator.

$$\frac{c}{c-2}$$

Expand

$$\frac{c}{c-2}$$

Short Solution Steps

Factor c^{2}+c-6.

$$\left(\frac{2}{c-2}+\frac{3c-21}{\left(c-2\right)\left(c+3\right)}+\frac{2c}{c+3}\right)\times \frac{c}{2c-5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of c-2 and \left(c-2\right)\left(c+3\right) is \left(c-2\right)\left(c+3\right). Multiply \frac{2}{c-2} times \frac{c+3}{c+3}.

$$\left(\frac{2\left(c+3\right)}{\left(c-2\right)\left(c+3\right)}+\frac{3c-21}{\left(c-2\right)\left(c+3\right)}+\frac{2c}{c+3}\right)\times \frac{c}{2c-5}$$

Since \frac{2\left(c+3\right)}{\left(c-2\right)\left(c+3\right)} and \frac{3c-21}{\left(c-2\right)\left(c+3\right)} have the same denominator, add them by adding their numerators.

$$\left(\frac{2\left(c+3\right)+3c-21}{\left(c-2\right)\left(c+3\right)}+\frac{2c}{c+3}\right)\times \frac{c}{2c-5}$$

Do the multiplications in 2\left(c+3\right)+3c-21.

$$\left(\frac{2c+6+3c-21}{\left(c-2\right)\left(c+3\right)}+\frac{2c}{c+3}\right)\times \frac{c}{2c-5}$$

Combine like terms in 2c+6+3c-21.

$$\left(\frac{5c-15}{\left(c-2\right)\left(c+3\right)}+\frac{2c}{c+3}\right)\times \frac{c}{2c-5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(c-2\right)\left(c+3\right) and c+3 is \left(c-2\right)\left(c+3\right). Multiply \frac{2c}{c+3} times \frac{c-2}{c-2}.

$$\left(\frac{5c-15}{\left(c-2\right)\left(c+3\right)}+\frac{2c\left(c-2\right)}{\left(c-2\right)\left(c+3\right)}\right)\times \frac{c}{2c-5}$$

Since \frac{5c-15}{\left(c-2\right)\left(c+3\right)} and \frac{2c\left(c-2\right)}{\left(c-2\right)\left(c+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{5c-15+2c\left(c-2\right)}{\left(c-2\right)\left(c+3\right)}\times \frac{c}{2c-5}$$

Do the multiplications in 5c-15+2c\left(c-2\right).

$$\frac{5c-15+2c^{2}-4c}{\left(c-2\right)\left(c+3\right)}\times \frac{c}{2c-5}$$

Combine like terms in 5c-15+2c^{2}-4c.

$$\frac{c-15+2c^{2}}{\left(c-2\right)\left(c+3\right)}\times \frac{c}{2c-5}$$

Factor the expressions that are not already factored in \frac{c-15+2c^{2}}{\left(c-2\right)\left(c+3\right)}.

$$\frac{\left(2c-5\right)\left(c+3\right)}{\left(c-2\right)\left(c+3\right)}\times \frac{c}{2c-5}$$

Cancel out c+3 in both numerator and denominator.

$$\frac{2c-5}{c-2}\times \frac{c}{2c-5}$$

Multiply \frac{2c-5}{c-2} times \frac{c}{2c-5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(2c-5\right)c}{\left(c-2\right)\left(2c-5\right)}$$

Cancel out 2c-5 in both numerator and denominator.

$$\frac{c}{c-2}$$