Solve a-32/a^2-a-12-16/a^2+3a+16/a^2-4a

$$\frac { a - 32 } { a ^ { 2 } - a - 12 } - \frac { 16 } { a ^ { 2 } + 3 a } + \frac { 16 } { a ^ { 2 } - 4 a }$$

Evaluate

$$\frac{a-28}{a\left(a+3\right)}$$

Short Solution Steps

Factor a^{2}-a-12. Factor a^{2}+3a.

$$\frac{a-32}{\left(a-4\right)\left(a+3\right)}-\frac{16}{a\left(a+3\right)}+\frac{16}{a^{2}-4a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-4\right)\left(a+3\right) and a\left(a+3\right) is a\left(a-4\right)\left(a+3\right). Multiply \frac{a-32}{\left(a-4\right)\left(a+3\right)} times \frac{a}{a}. Multiply \frac{16}{a\left(a+3\right)} times \frac{a-4}{a-4}.

$$\frac{\left(a-32\right)a}{a\left(a-4\right)\left(a+3\right)}-\frac{16\left(a-4\right)}{a\left(a-4\right)\left(a+3\right)}+\frac{16}{a^{2}-4a}$$

Since \frac{\left(a-32\right)a}{a\left(a-4\right)\left(a+3\right)} and \frac{16\left(a-4\right)}{a\left(a-4\right)\left(a+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a-32\right)a-16\left(a-4\right)}{a\left(a-4\right)\left(a+3\right)}+\frac{16}{a^{2}-4a}$$

Do the multiplications in \left(a-32\right)a-16\left(a-4\right).

$$\frac{a^{2}-32a-16a+64}{a\left(a-4\right)\left(a+3\right)}+\frac{16}{a^{2}-4a}$$

Combine like terms in a^{2}-32a-16a+64.

$$\frac{a^{2}-48a+64}{a\left(a-4\right)\left(a+3\right)}+\frac{16}{a^{2}-4a}$$

Factor a^{2}-4a.

$$\frac{a^{2}-48a+64}{a\left(a-4\right)\left(a+3\right)}+\frac{16}{a\left(a-4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a-4\right)\left(a+3\right) and a\left(a-4\right) is a\left(a-4\right)\left(a+3\right). Multiply \frac{16}{a\left(a-4\right)} times \frac{a+3}{a+3}.

$$\frac{a^{2}-48a+64}{a\left(a-4\right)\left(a+3\right)}+\frac{16\left(a+3\right)}{a\left(a-4\right)\left(a+3\right)}$$

Since \frac{a^{2}-48a+64}{a\left(a-4\right)\left(a+3\right)} and \frac{16\left(a+3\right)}{a\left(a-4\right)\left(a+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{a^{2}-48a+64+16\left(a+3\right)}{a\left(a-4\right)\left(a+3\right)}$$

Do the multiplications in a^{2}-48a+64+16\left(a+3\right).

$$\frac{a^{2}-48a+64+16a+48}{a\left(a-4\right)\left(a+3\right)}$$

Combine like terms in a^{2}-48a+64+16a+48.

$$\frac{a^{2}-32a+112}{a\left(a-4\right)\left(a+3\right)}$$

Factor the expressions that are not already factored in \frac{a^{2}-32a+112}{a\left(a-4\right)\left(a+3\right)}.

$$\frac{\left(a-28\right)\left(a-4\right)}{a\left(a-4\right)\left(a+3\right)}$$

Cancel out a-4 in both numerator and denominator.

$$\frac{a-28}{a\left(a+3\right)}$$

Expand a\left(a+3\right).

$$\frac{a-28}{a^{2}+3a}$$

Expand

$$\frac{a-28}{a\left(a+3\right)}$$

Short Solution Steps

Factor a^{2}-a-12. Factor a^{2}+3a.

$$\frac{a-32}{\left(a-4\right)\left(a+3\right)}-\frac{16}{a\left(a+3\right)}+\frac{16}{a^{2}-4a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-4\right)\left(a+3\right) and a\left(a+3\right) is a\left(a-4\right)\left(a+3\right). Multiply \frac{a-32}{\left(a-4\right)\left(a+3\right)} times \frac{a}{a}. Multiply \frac{16}{a\left(a+3\right)} times \frac{a-4}{a-4}.

$$\frac{\left(a-32\right)a}{a\left(a-4\right)\left(a+3\right)}-\frac{16\left(a-4\right)}{a\left(a-4\right)\left(a+3\right)}+\frac{16}{a^{2}-4a}$$

Since \frac{\left(a-32\right)a}{a\left(a-4\right)\left(a+3\right)} and \frac{16\left(a-4\right)}{a\left(a-4\right)\left(a+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a-32\right)a-16\left(a-4\right)}{a\left(a-4\right)\left(a+3\right)}+\frac{16}{a^{2}-4a}$$

Do the multiplications in \left(a-32\right)a-16\left(a-4\right).

$$\frac{a^{2}-32a-16a+64}{a\left(a-4\right)\left(a+3\right)}+\frac{16}{a^{2}-4a}$$

Combine like terms in a^{2}-32a-16a+64.

$$\frac{a^{2}-48a+64}{a\left(a-4\right)\left(a+3\right)}+\frac{16}{a^{2}-4a}$$

Factor a^{2}-4a.

$$\frac{a^{2}-48a+64}{a\left(a-4\right)\left(a+3\right)}+\frac{16}{a\left(a-4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a-4\right)\left(a+3\right) and a\left(a-4\right) is a\left(a-4\right)\left(a+3\right). Multiply \frac{16}{a\left(a-4\right)} times \frac{a+3}{a+3}.

$$\frac{a^{2}-48a+64}{a\left(a-4\right)\left(a+3\right)}+\frac{16\left(a+3\right)}{a\left(a-4\right)\left(a+3\right)}$$

Since \frac{a^{2}-48a+64}{a\left(a-4\right)\left(a+3\right)} and \frac{16\left(a+3\right)}{a\left(a-4\right)\left(a+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{a^{2}-48a+64+16\left(a+3\right)}{a\left(a-4\right)\left(a+3\right)}$$

Do the multiplications in a^{2}-48a+64+16\left(a+3\right).

$$\frac{a^{2}-48a+64+16a+48}{a\left(a-4\right)\left(a+3\right)}$$

Combine like terms in a^{2}-48a+64+16a+48.

$$\frac{a^{2}-32a+112}{a\left(a-4\right)\left(a+3\right)}$$

Factor the expressions that are not already factored in \frac{a^{2}-32a+112}{a\left(a-4\right)\left(a+3\right)}.

$$\frac{\left(a-28\right)\left(a-4\right)}{a\left(a-4\right)\left(a+3\right)}$$

Cancel out a-4 in both numerator and denominator.

$$\frac{a-28}{a\left(a+3\right)}$$

Expand a\left(a+3\right).

$$\frac{a-28}{a^{2}+3a}$$