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

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

Calcular

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

Pasos de solución curta

Factoriza 12c-c^{2}.

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

Para sumar ou restar expresións, expándeas para facer que os seus denominadores sexan iguais. O mínimo común múltiplo de \left(12-c\right)^{2} e c\left(-c+12\right) é c\left(-c+12\right)\left(-c+12\right)^{2}. Multiplica \frac{c+12}{\left(12-c\right)^{2}} por \frac{c\left(-c+12\right)}{c\left(-c+12\right)}. Multiplica \frac{12}{c\left(-c+12\right)} por \frac{\left(-c+12\right)^{2}}{\left(-c+12\right)^{2}}.

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

Dado que \frac{\left(c+12\right)c\left(-c+12\right)}{c\left(-c+12\right)\left(-c+12\right)^{2}} e \frac{12\left(-c+12\right)^{2}}{c\left(-c+12\right)\left(-c+12\right)^{2}} teñen o mesmo denominador, súmaos mediante a suma dos seus numeradores.

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

Fai as multiplicacións en \left(c+12\right)c\left(-c+12\right)+12\left(-c+12\right)^{2}.

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

Combina como termos en -c^{3}+12c^{2}-12c^{2}+144c+12c^{2}-288c+1728.

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

Factoriza as expresións que aínda non o están en \frac{-c^{3}+12c^{2}-144c+1728}{c\left(-c+12\right)\left(-c+12\right)^{2}}.

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

Anula -c+12 no numerador e no denominador.

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

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

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

Expandir

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

Pasos de solución curta

Factoriza 12c-c^{2}.

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

Para sumar ou restar expresións, expándeas para facer que os seus denominadores sexan iguais. O mínimo común múltiplo de \left(12-c\right)^{2} e c\left(-c+12\right) é c\left(-c+12\right)\left(-c+12\right)^{2}. Multiplica \frac{c+12}{\left(12-c\right)^{2}} por \frac{c\left(-c+12\right)}{c\left(-c+12\right)}. Multiplica \frac{12}{c\left(-c+12\right)} por \frac{\left(-c+12\right)^{2}}{\left(-c+12\right)^{2}}.

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

Dado que \frac{\left(c+12\right)c\left(-c+12\right)}{c\left(-c+12\right)\left(-c+12\right)^{2}} e \frac{12\left(-c+12\right)^{2}}{c\left(-c+12\right)\left(-c+12\right)^{2}} teñen o mesmo denominador, súmaos mediante a suma dos seus numeradores.

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

Fai as multiplicacións en \left(c+12\right)c\left(-c+12\right)+12\left(-c+12\right)^{2}.

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

Combina como termos en -c^{3}+12c^{2}-12c^{2}+144c+12c^{2}-288c+1728.

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

Factoriza as expresións que aínda non o están en \frac{-c^{3}+12c^{2}-144c+1728}{c\left(-c+12\right)\left(-c+12\right)^{2}}.

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

Anula -c+12 no numerador e no denominador.

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

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

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