Solve frac{a^{2}+ac}{a^{2}c-c^{3}}-frac{a^{2}-c^{2}}{a^{2}c-2ac^2+c^3}+2c/c^2-a^2-3/a+c

$$\frac { a ^ { 2 } + a c } { a ^ { 2 } c - c ^ { 3 } } - \frac { a ^ { 2 } - c ^ { 2 } } { a ^ { 2 } c - 2 a c ^ { 2 } + c ^ { 3 } } + \frac { 2 c } { c ^ { 2 } - a ^ { 2 } } - \frac { 3 } { a + c }$$

Evaluate

$$\frac{4a}{c^{2}-a^{2}}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{2}+ac}{a^{2}c-c^{3}}.

$$\frac{a\left(a+c\right)}{c\left(a+c\right)\left(a-c\right)}-\frac{a^{2}-c^{2}}{a^{2}c-2ac^{2}+c^{3}}+\frac{2c}{c^{2}-a^{2}}-\frac{3}{a+c}$$

Cancel out a+c in both numerator and denominator.

$$\frac{a}{c\left(a-c\right)}-\frac{a^{2}-c^{2}}{a^{2}c-2ac^{2}+c^{3}}+\frac{2c}{c^{2}-a^{2}}-\frac{3}{a+c}$$

Factor the expressions that are not already factored in \frac{a^{2}-c^{2}}{a^{2}c-2ac^{2}+c^{3}}.

$$\frac{a}{c\left(a-c\right)}-\frac{\left(a+c\right)\left(a-c\right)}{c\left(a-c\right)^{2}}+\frac{2c}{c^{2}-a^{2}}-\frac{3}{a+c}$$

Cancel out a-c in both numerator and denominator.

$$\frac{a}{c\left(a-c\right)}-\frac{a+c}{c\left(a-c\right)}+\frac{2c}{c^{2}-a^{2}}-\frac{3}{a+c}$$

Since \frac{a}{c\left(a-c\right)} and \frac{a+c}{c\left(a-c\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a-\left(a+c\right)}{c\left(a-c\right)}+\frac{2c}{c^{2}-a^{2}}-\frac{3}{a+c}$$

Do the multiplications in a-\left(a+c\right).

$$\frac{a-a-c}{c\left(a-c\right)}+\frac{2c}{c^{2}-a^{2}}-\frac{3}{a+c}$$

Combine like terms in a-a-c.

$$\frac{-c}{c\left(a-c\right)}+\frac{2c}{c^{2}-a^{2}}-\frac{3}{a+c}$$

Cancel out c in both numerator and denominator.

$$\frac{-1}{a-c}+\frac{2c}{c^{2}-a^{2}}-\frac{3}{a+c}$$

Factor c^{2}-a^{2}.

$$\frac{-1}{a-c}+\frac{2c}{\left(a+c\right)\left(-a+c\right)}-\frac{3}{a+c}$$

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

$$\frac{-\left(-1\right)\left(a+c\right)}{\left(a+c\right)\left(-a+c\right)}+\frac{2c}{\left(a+c\right)\left(-a+c\right)}-\frac{3}{a+c}$$

Since \frac{-\left(-1\right)\left(a+c\right)}{\left(a+c\right)\left(-a+c\right)} and \frac{2c}{\left(a+c\right)\left(-a+c\right)} have the same denominator, add them by adding their numerators.

$$\frac{-\left(-1\right)\left(a+c\right)+2c}{\left(a+c\right)\left(-a+c\right)}-\frac{3}{a+c}$$

Do the multiplications in -\left(-1\right)\left(a+c\right)+2c.

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

Combine like terms in a+c+2c.

$$\frac{a+3c}{\left(a+c\right)\left(-a+c\right)}-\frac{3}{a+c}$$

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

$$\frac{a+3c}{\left(a+c\right)\left(-a+c\right)}-\frac{3\left(-a+c\right)}{\left(a+c\right)\left(-a+c\right)}$$

Since \frac{a+3c}{\left(a+c\right)\left(-a+c\right)} and \frac{3\left(-a+c\right)}{\left(a+c\right)\left(-a+c\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a+3c-3\left(-a+c\right)}{\left(a+c\right)\left(-a+c\right)}$$

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

$$\frac{a+3c+3a-3c}{\left(a+c\right)\left(-a+c\right)}$$

Combine like terms in a+3c+3a-3c.

$$\frac{4a}{\left(a+c\right)\left(-a+c\right)}$$

Expand \left(a+c\right)\left(-a+c\right).

$$\frac{4a}{-a^{2}+c^{2}}$$

Expand

$$\frac{4a}{c^{2}-a^{2}}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{2}+ac}{a^{2}c-c^{3}}.

$$\frac{a\left(a+c\right)}{c\left(a+c\right)\left(a-c\right)}-\frac{a^{2}-c^{2}}{a^{2}c-2ac^{2}+c^{3}}+\frac{2c}{c^{2}-a^{2}}-\frac{3}{a+c}$$

Cancel out a+c in both numerator and denominator.

$$\frac{a}{c\left(a-c\right)}-\frac{a^{2}-c^{2}}{a^{2}c-2ac^{2}+c^{3}}+\frac{2c}{c^{2}-a^{2}}-\frac{3}{a+c}$$

Factor the expressions that are not already factored in \frac{a^{2}-c^{2}}{a^{2}c-2ac^{2}+c^{3}}.

$$\frac{a}{c\left(a-c\right)}-\frac{\left(a+c\right)\left(a-c\right)}{c\left(a-c\right)^{2}}+\frac{2c}{c^{2}-a^{2}}-\frac{3}{a+c}$$

Cancel out a-c in both numerator and denominator.

$$\frac{a}{c\left(a-c\right)}-\frac{a+c}{c\left(a-c\right)}+\frac{2c}{c^{2}-a^{2}}-\frac{3}{a+c}$$

Since \frac{a}{c\left(a-c\right)} and \frac{a+c}{c\left(a-c\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a-\left(a+c\right)}{c\left(a-c\right)}+\frac{2c}{c^{2}-a^{2}}-\frac{3}{a+c}$$

Do the multiplications in a-\left(a+c\right).

$$\frac{a-a-c}{c\left(a-c\right)}+\frac{2c}{c^{2}-a^{2}}-\frac{3}{a+c}$$

Combine like terms in a-a-c.

$$\frac{-c}{c\left(a-c\right)}+\frac{2c}{c^{2}-a^{2}}-\frac{3}{a+c}$$

Cancel out c in both numerator and denominator.

$$\frac{-1}{a-c}+\frac{2c}{c^{2}-a^{2}}-\frac{3}{a+c}$$

Factor c^{2}-a^{2}.

$$\frac{-1}{a-c}+\frac{2c}{\left(a+c\right)\left(-a+c\right)}-\frac{3}{a+c}$$

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

$$\frac{-\left(-1\right)\left(a+c\right)}{\left(a+c\right)\left(-a+c\right)}+\frac{2c}{\left(a+c\right)\left(-a+c\right)}-\frac{3}{a+c}$$

Since \frac{-\left(-1\right)\left(a+c\right)}{\left(a+c\right)\left(-a+c\right)} and \frac{2c}{\left(a+c\right)\left(-a+c\right)} have the same denominator, add them by adding their numerators.

$$\frac{-\left(-1\right)\left(a+c\right)+2c}{\left(a+c\right)\left(-a+c\right)}-\frac{3}{a+c}$$

Do the multiplications in -\left(-1\right)\left(a+c\right)+2c.

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

Combine like terms in a+c+2c.

$$\frac{a+3c}{\left(a+c\right)\left(-a+c\right)}-\frac{3}{a+c}$$

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

$$\frac{a+3c}{\left(a+c\right)\left(-a+c\right)}-\frac{3\left(-a+c\right)}{\left(a+c\right)\left(-a+c\right)}$$

Since \frac{a+3c}{\left(a+c\right)\left(-a+c\right)} and \frac{3\left(-a+c\right)}{\left(a+c\right)\left(-a+c\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a+3c-3\left(-a+c\right)}{\left(a+c\right)\left(-a+c\right)}$$

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

$$\frac{a+3c+3a-3c}{\left(a+c\right)\left(-a+c\right)}$$

Combine like terms in a+3c+3a-3c.

$$\frac{4a}{\left(a+c\right)\left(-a+c\right)}$$

Expand \left(a+c\right)\left(-a+c\right).

$$\frac{4a}{-a^{2}+c^{2}}$$