Resolver x^2/(4-x)(1-x)=13*10^9

$$\frac { x ^ { 2 } } { ( 4 - x ) ( 1 - x ) } = 13 \times 10 ^ { 9 }$$

Resolver x

$$x = \frac{20000 \sqrt{950625000130} + 32500000000}{12999999999} \approx 4$$

$$x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999}\approx 1$$

Pasos que usan a fórmula cadrática

A variable x non pode ser igual a ningún dos valores 1,4 porque a división entre cero non está definida. Multiplica ambos lados da ecuación por \left(x-4\right)\left(x-1\right).

$$x^{2}=13\times 10^{9}\left(x-4\right)\left(x-1\right)$$

Calcula 10 á potencia de 9 e obtén 1000000000.

$$x^{2}=13\times 1000000000\left(x-4\right)\left(x-1\right)$$

Multiplica 13 e 1000000000 para obter 13000000000.

$$x^{2}=13000000000\left(x-4\right)\left(x-1\right)$$

Usa a propiedade distributiva para multiplicar 13000000000 por x-4.

$$x^{2}=\left(13000000000x-52000000000\right)\left(x-1\right)$$

Usa a propiedade distributiva para multiplicar 13000000000x-52000000000 por x-1 e combina os termos semellantes.

$$x^{2}=13000000000x^{2}-65000000000x+52000000000$$

Resta 13000000000x^{2} en ambos lados.

$$x^{2}-13000000000x^{2}=-65000000000x+52000000000$$

Combina x^{2} e -13000000000x^{2} para obter -12999999999x^{2}.

$$-12999999999x^{2}=-65000000000x+52000000000$$

Engadir 65000000000x en ambos lados.

$$-12999999999x^{2}[phone]x=52000000000$$

Resta 52000000000 en ambos lados.

$$-12999999999x^{2}[phone]x-52000000000=0$$

Esta ecuación ten unha forma estándar: ax^{2}+bx+c=0. Substitúe a por -12999999999, b por 65000000000 e c por -52000000000 na fórmula cadrática, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-65000000000±\sqrt{65000000000^{2}-4\left(-12999999999\right)\left(-52000000000\right)}}{2\left(-12999999999\right)}$$

Eleva 65000000000 ao cadrado.

$$x=\frac{-65000000000±\sqrt{4225000000000000000000-4\left(-12999999999\right)\left(-52000000000\right)}}{2\left(-12999999999\right)}$$

Multiplica -4 por -12999999999.

$$x=\frac{-65000000000±\sqrt{4225000000000000000000+51999999996\left(-52000000000\right)}}{2\left(-12999999999\right)}$$

Multiplica 51999999996 por -52000000000.

$$x=\frac{-65000000000±\sqrt{4225000000000000000000-2703999999792000000000}}{2\left(-12999999999\right)}$$

Suma 4225000000000000000000 a -2703999999792000000000.

$$x=\frac{-65000000000±\sqrt{1521000000208000000000}}{2\left(-12999999999\right)}$$

Obtén a raíz cadrada de 1521000000208000000000.

$$x=\frac{-65000000000±40000\sqrt{950625000130}}{2\left(-12999999999\right)}$$

Multiplica 2 por -12999999999.

$$x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998}$$

Agora resolve a ecuación x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998} se ± é máis. Suma -65000000000 a 40000\sqrt{950625000130}.

$$x=\frac{40000\sqrt{950625000130}-65000000000}{-25999999998}$$

Divide -65000000000+40000\sqrt{950625000130} entre -25999999998.

$$x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999}$$

Agora resolve a ecuación x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998} se ± é menos. Resta 40000\sqrt{950625000130} de -65000000000.

$$x=\frac{-40000\sqrt{950625000130}-65000000000}{-25999999998}$$

Divide -65000000000-40000\sqrt{950625000130} entre -25999999998.

$$x=\frac{20000\sqrt{950625000130}[phone]}{12999999999}$$

A ecuación está resolta.

$$x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999} x=\frac{20000\sqrt{950625000130}[phone]}{12999999999}$$

Pasos para completar o cadrado

A variable x non pode ser igual a ningún dos valores 1,4 porque a división entre cero non está definida. Multiplica ambos lados da ecuación por \left(x-4\right)\left(x-1\right).

$$x^{2}=13\times 10^{9}\left(x-4\right)\left(x-1\right)$$

Calcula 10 á potencia de 9 e obtén 1000000000.

$$x^{2}=13\times 1000000000\left(x-4\right)\left(x-1\right)$$

Multiplica 13 e 1000000000 para obter 13000000000.

$$x^{2}=13000000000\left(x-4\right)\left(x-1\right)$$

Usa a propiedade distributiva para multiplicar 13000000000 por x-4.

$$x^{2}=\left(13000000000x-52000000000\right)\left(x-1\right)$$

Usa a propiedade distributiva para multiplicar 13000000000x-52000000000 por x-1 e combina os termos semellantes.

$$x^{2}=13000000000x^{2}-65000000000x+52000000000$$

Resta 13000000000x^{2} en ambos lados.

$$x^{2}-13000000000x^{2}=-65000000000x+52000000000$$

Combina x^{2} e -13000000000x^{2} para obter -12999999999x^{2}.

$$-12999999999x^{2}=-65000000000x+52000000000$$

Engadir 65000000000x en ambos lados.

$$-12999999999x^{2}[phone]x=52000000000$$

Divide ambos lados entre -12999999999.

$$\frac{-12999999999x^{2}[phone]x}{-12999999999}=\frac{52000000000}{-12999999999}$$

A división entre -12999999999 desfai a multiplicación por -12999999999.

$$x^{2}+\frac{65000000000}{-12999999999}x=\frac{52000000000}{-12999999999}$$

Divide 65000000000 entre -12999999999.

$$x^{2}-\frac{65000000000}{12999999999}x=\frac{52000000000}{-12999999999}$$

Divide 52000000000 entre -12999999999.

$$x^{2}-\frac{65000000000}{12999999999}x=-\frac{52000000000}{12999999999}$$

Divide -\frac{65000000000}{12999999999}, o coeficiente do termo x, entre 2 para obter -\frac{32500000000}{12999999999}. Despois, suma o cadrado de -\frac{32500000000}{12999999999} en ambos lados da ecuación. Este paso converte o lado esquerdo da ecuación nun cadrado perfecto.

$$x^{2}-\frac{65000000000}{12999999999}x+\left(-\frac{32500000000}{12999999999}\right)^{2}=-\frac{52000000000}{12999999999}+\left(-\frac{32500000000}{12999999999}\right)^{2}$$

Eleva -\frac{32500000000}{12999999999} ao cadrado mediante a elevación ao cadrado do numerador e do denominador da fracción.

$$x^{2}-\frac{65000000000}{12999999999}x+\frac{1056250000000000000000}{168999999974000000001}=-\frac{52000000000}{12999999999}+\frac{1056250000000000000000}{168999999974000000001}$$

Suma -\frac{52000000000}{12999999999} a \frac{1056250000000000000000}{168999999974000000001} mediante a busca dun denominador común e a suma dos numeradores. Despois, se é posible, reduce a fracción aos termos máis baixos.

$$x^{2}-\frac{65000000000}{12999999999}x+\frac{1056250000000000000000}{168999999974000000001}=\frac{380250000052000000000}{168999999974000000001}$$

Factoriza x^{2}-\frac{65000000000}{12999999999}x+\frac{1056250000000000000000}{168999999974000000001}. En xeral, cando x^{2}+bx+c é un cadrado perfecto, sempre se pode factorizar como \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{32500000000}{12999999999}\right)^{2}=\frac{380250000052000000000}{168999999974000000001}$$

Obtén a raíz cadrada de ambos lados da ecuación.

$$\sqrt{\left(x-\frac{32500000000}{12999999999}\right)^{2}}=\sqrt{\frac{380250000052000000000}{168999999974000000001}}$$

Simplifica.

$$x-\frac{32500000000}{12999999999}=\frac{20000\sqrt{950625000130}}{12999999999} x-\frac{32500000000}{12999999999}=-\frac{20000\sqrt{950625000130}}{12999999999}$$

Suma \frac{32500000000}{12999999999} en ambos lados da ecuación.

$$x=\frac{20000\sqrt{950625000130}[phone]}{12999999999} x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999}$$