Resolver 20^2+left(1.5tright)^2-left(12+1.5tright)^2/60t=frac{34^2+{left(1.5tright)}^2-{left(30+1.5tright)}^2}{-102t}

$$\frac{ { 20 }^{ 2 } + { \left(1.5t \right) }^{ 2 } - { \left(12+1.5t \right) }^{ 2 } }{ 60t } = \frac{ { 34 }^{ 2 } + { \left(1.5t \right) }^{ 2 } - { \left(30+1.5t \right) }^{ 2 } }{ -102t }$$

Resolver t

$$t = \frac{32}{7} = 4\frac{4}{7} \approx 4.571428571$$

Pasos para resolver unha ecuación lineal

A variable t non pode ser igual a 0 porque a división entre cero non está definida. Multiplica ambos lados da ecuación por 1020t, o mínimo común denominador de 60t,-102t.

$$17\left(20^{2}+\left(1.5t\right)^{2}-\left(12+1.5t\right)^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Calcula 20 á potencia de 2 e obtén 400.

$$17\left(400+\left(1.5t\right)^{2}-\left(12+1.5t\right)^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Expande \left(1.5t\right)^{2}.

$$17\left(400+1.5^{2}t^{2}-\left(12+1.5t\right)^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Calcula 1.5 á potencia de 2 e obtén 2.25.

$$17\left(400+2.25t^{2}-\left(12+1.5t\right)^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Usar teorema binomial \left(a+b\right)^{2}=a^{2}+2ab+b^{2} para expandir \left(12+1.5t\right)^{2}.

$$17\left(400+2.25t^{2}-\left(144+36t+2.25t^{2}\right)\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Para calcular o oposto de 144+36t+2.25t^{2}, calcula o oposto de cada termo.

$$17\left(400+2.25t^{2}-144-36t-2.25t^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Resta 144 de 400 para obter 256.

$$17\left(256+2.25t^{2}-36t-2.25t^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Combina 2.25t^{2} e -2.25t^{2} para obter 0.

$$17\left(256-36t\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Usa a propiedade distributiva para multiplicar 17 por 256-36t.

$$4352-612t=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Calcula 34 á potencia de 2 e obtén 1156.

$$4352-612t=-10\left(1156+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Expande \left(1.5t\right)^{2}.

$$4352-612t=-10\left(1156+1.5^{2}t^{2}-\left(30+1.5t\right)^{2}\right)$$

Calcula 1.5 á potencia de 2 e obtén 2.25.

$$4352-612t=-10\left(1156+2.25t^{2}-\left(30+1.5t\right)^{2}\right)$$

Usar teorema binomial \left(a+b\right)^{2}=a^{2}+2ab+b^{2} para expandir \left(30+1.5t\right)^{2}.

$$4352-612t=-10\left(1156+2.25t^{2}-\left(900+90t+2.25t^{2}\right)\right)$$

Para calcular o oposto de 900+90t+2.25t^{2}, calcula o oposto de cada termo.

$$4352-612t=-10\left(1156+2.25t^{2}-900-90t-2.25t^{2}\right)$$

Resta 900 de 1156 para obter 256.

$$4352-612t=-10\left(256+2.25t^{2}-90t-2.25t^{2}\right)$$

Combina 2.25t^{2} e -2.25t^{2} para obter 0.

$$4352-612t=-10\left(256-90t\right)$$

Usa a propiedade distributiva para multiplicar -10 por 256-90t.

$$4352-612t=-2560+900t$$

Resta 900t en ambos lados.

$$4352-612t-900t=-2560$$

Combina -612t e -900t para obter -1512t.

$$4352-1512t=-2560$$

Resta 4352 en ambos lados.

$$-1512t=-2560-4352$$

Resta 4352 de -2560 para obter -6912.

$$-1512t=-6912$$

Divide ambos lados entre -1512.

$$t=\frac{-6912}{-1512}$$

Reduce a fracción \frac{-6912}{-1512} a termos máis baixos extraendo e cancelando -216.

$$t=\frac{32}{7}$$