Solve c+12/(12-c)^2+12/12c-c^2

$$\frac { c + 12 } { ( 12 - c ) ^ { 2 } } + \frac { 12 } { 12 c - c ^ { 2 } }$$

Evaluate

$$\frac{c^{2}+144}{c\left(12-c\right)^{2}}$$

Short Solution Steps

Factor 12c-c^{2}.

$$\frac{c+12}{\left(12-c\right)^{2}}+\frac{12}{c\left(-c+12\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(12-c\right)^{2} and c\left(-c+12\right) is c\left(-c+12\right)\left(-c+12\right)^{2}. Multiply \frac{c+12}{\left(12-c\right)^{2}} times \frac{c\left(-c+12\right)}{c\left(-c+12\right)}. Multiply \frac{12}{c\left(-c+12\right)} times \frac{\left(-c+12\right)^{2}}{\left(-c+12\right)^{2}}.

$$\frac{\left(c+12\right)c\left(-c+12\right)}{c\left(-c+12\right)\left(-c+12\right)^{2}}+\frac{12\left(-c+12\right)^{2}}{c\left(-c+12\right)\left(-c+12\right)^{2}}$$

Since \frac{\left(c+12\right)c\left(-c+12\right)}{c\left(-c+12\right)\left(-c+12\right)^{2}} and \frac{12\left(-c+12\right)^{2}}{c\left(-c+12\right)\left(-c+12\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(c+12\right)c\left(-c+12\right)+12\left(-c+12\right)^{2}}{c\left(-c+12\right)\left(-c+12\right)^{2}}$$

Do the multiplications in \left(c+12\right)c\left(-c+12\right)+12\left(-c+12\right)^{2}.

$$\frac{-c^{3}+12c^{2}-12c^{2}+144c+12c^{2}-288c+1728}{c\left(-c+12\right)\left(-c+12\right)^{2}}$$

Combine like terms in -c^{3}+12c^{2}-12c^{2}+144c+12c^{2}-288c+1728.

$$\frac{-c^{3}+12c^{2}-144c+1728}{c\left(-c+12\right)\left(-c+12\right)^{2}}$$

Factor the expressions that are not already factored in \frac{-c^{3}+12c^{2}-144c+1728}{c\left(-c+12\right)\left(-c+12\right)^{2}}.

$$\frac{\left(-c+12\right)\left(c^{2}+144\right)}{c\left(-c+12\right)\left(-c+12\right)^{2}}$$

Cancel out -c+12 in both numerator and denominator.

$$\frac{c^{2}+144}{c\left(-c+12\right)^{2}}$$

Expand c\left(-c+12\right)^{2}.

$$\frac{c^{2}+144}{c^{3}-24c^{2}+144c}$$

Expand

$$\frac{c^{2}+144}{c\left(c-12\right)^{2}}$$

Short Solution Steps

Factor 12c-c^{2}.

$$\frac{c+12}{\left(12-c\right)^{2}}+\frac{12}{c\left(-c+12\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(12-c\right)^{2} and c\left(-c+12\right) is c\left(-c+12\right)\left(-c+12\right)^{2}. Multiply \frac{c+12}{\left(12-c\right)^{2}} times \frac{c\left(-c+12\right)}{c\left(-c+12\right)}. Multiply \frac{12}{c\left(-c+12\right)} times \frac{\left(-c+12\right)^{2}}{\left(-c+12\right)^{2}}.

$$\frac{\left(c+12\right)c\left(-c+12\right)}{c\left(-c+12\right)\left(-c+12\right)^{2}}+\frac{12\left(-c+12\right)^{2}}{c\left(-c+12\right)\left(-c+12\right)^{2}}$$

Since \frac{\left(c+12\right)c\left(-c+12\right)}{c\left(-c+12\right)\left(-c+12\right)^{2}} and \frac{12\left(-c+12\right)^{2}}{c\left(-c+12\right)\left(-c+12\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(c+12\right)c\left(-c+12\right)+12\left(-c+12\right)^{2}}{c\left(-c+12\right)\left(-c+12\right)^{2}}$$

Do the multiplications in \left(c+12\right)c\left(-c+12\right)+12\left(-c+12\right)^{2}.

$$\frac{-c^{3}+12c^{2}-12c^{2}+144c+12c^{2}-288c+1728}{c\left(-c+12\right)\left(-c+12\right)^{2}}$$

Combine like terms in -c^{3}+12c^{2}-12c^{2}+144c+12c^{2}-288c+1728.

$$\frac{-c^{3}+12c^{2}-144c+1728}{c\left(-c+12\right)\left(-c+12\right)^{2}}$$

Factor the expressions that are not already factored in \frac{-c^{3}+12c^{2}-144c+1728}{c\left(-c+12\right)\left(-c+12\right)^{2}}.

$$\frac{\left(-c+12\right)\left(c^{2}+144\right)}{c\left(-c+12\right)\left(-c+12\right)^{2}}$$

Cancel out -c+12 in both numerator and denominator.

$$\frac{c^{2}+144}{c\left(-c+12\right)^{2}}$$

Expand c\left(-c+12\right)^{2}.

$$\frac{c^{2}+144}{c^{3}-24c^{2}+144c}$$