Resolver 3-xsqrt{6}-xsqrt{2}-x^2+x^2sqrt{6}+2x^2+2+4x+2=0

$$3 - x \sqrt { 6 } - x \sqrt { 2 } - x ^ { 2 } + x ^ { 2 } \sqrt { 6 } + 2 x ^ { 2 } + 2 + 4 x + 2 = 0$$

Resolver x (complex solution)

$$x=\frac{\left(\sqrt{6}-1\right)\left(\sqrt{2}+\sqrt{6}-4+2i\sqrt{2\sqrt{2}+9\sqrt{6}+1-\sqrt{3}}\right)}{10}\approx -0.019756066+1.424392972i$$

$$x=\frac{\left(\sqrt{6}-1\right)\left(-2i\sqrt{2\sqrt{2}+9\sqrt{6}+1-\sqrt{3}}+\sqrt{2}+\sqrt{6}-4\right)}{10}\approx -0.019756066-1.424392972i$$

Pasos que usan a fórmula cadrática

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

$$3-x\sqrt{6}-x\sqrt{2}+x^{2}+x^{2}\sqrt{6}+2+4x+2=0$$

Suma 2 e 2 para obter 4.

$$3-x\sqrt{6}-x\sqrt{2}+x^{2}+x^{2}\sqrt{6}+4+4x=0$$

Suma 3 e 4 para obter 7.

$$7-x\sqrt{6}-x\sqrt{2}+x^{2}+x^{2}\sqrt{6}+4x=0$$

Combina todos os termos que conteñan x.

$$7+\left(-\sqrt{6}-\sqrt{2}+4\right)x+\left(1+\sqrt{6}\right)x^{2}=0$$

Todas as ecuacións na forma ax^{2}+bx+c=0 pódense resolver coa fórmula cadrática: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. A fórmula cadrática fornece dúas solucións, unha cando ± é suma e outra cando é resta.

$$\left(\sqrt{6}+1\right)x^{2}+\left(4-\sqrt{2}-\sqrt{6}\right)x+7=0$$

Esta ecuación ten unha forma estándar: ax^{2}+bx+c=0. Substitúe a por 1+\sqrt{6}, b por -\sqrt{6}-\sqrt{2}+4 e c por 7 na fórmula cadrática, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(4-\sqrt{2}-\sqrt{6}\right)±\sqrt{\left(4-\sqrt{2}-\sqrt{6}\right)^{2}-4\left(\sqrt{6}+1\right)\times 7}}{2\left(\sqrt{6}+1\right)}$$

Eleva -\sqrt{6}-\sqrt{2}+4 ao cadrado.

$$x=\frac{-\left(4-\sqrt{2}-\sqrt{6}\right)±\sqrt{4\sqrt{3}+24-8\sqrt{2}-8\sqrt{6}-4\left(\sqrt{6}+1\right)\times 7}}{2\left(\sqrt{6}+1\right)}$$

Multiplica -4 por 1+\sqrt{6}.

$$x=\frac{-\left(4-\sqrt{2}-\sqrt{6}\right)±\sqrt{4\sqrt{3}+24-8\sqrt{2}-8\sqrt{6}+\left(-4\sqrt{6}-4\right)\times 7}}{2\left(\sqrt{6}+1\right)}$$

Multiplica -4-4\sqrt{6} por 7.

$$x=\frac{-\left(4-\sqrt{2}-\sqrt{6}\right)±\sqrt{4\sqrt{3}+24-8\sqrt{2}-8\sqrt{6}-28\sqrt{6}-28}}{2\left(\sqrt{6}+1\right)}$$

Suma 24-8\sqrt{6}+4\sqrt{3}-8\sqrt{2} a -28-28\sqrt{6}.

$$x=\frac{-\left(4-\sqrt{2}-\sqrt{6}\right)±\sqrt{4\sqrt{3}-8\sqrt{2}-36\sqrt{6}-4}}{2\left(\sqrt{6}+1\right)}$$

Obtén a raíz cadrada de -4-36\sqrt{6}+4\sqrt{3}-8\sqrt{2}.

$$x=\frac{-\left(4-\sqrt{2}-\sqrt{6}\right)±2i\sqrt{2\sqrt{2}+9\sqrt{6}+1-\sqrt{3}}}{2\left(\sqrt{6}+1\right)}$$

Multiplica 2 por 1+\sqrt{6}.

$$x=\frac{\sqrt{2}+\sqrt{6}-4±2i\sqrt{2\sqrt{2}+9\sqrt{6}+1-\sqrt{3}}}{2\sqrt{6}+2}$$

Agora resolve a ecuación x=\frac{\sqrt{2}+\sqrt{6}-4±2i\sqrt{2\sqrt{2}+9\sqrt{6}+1-\sqrt{3}}}{2\sqrt{6}+2} se ± é máis. Suma \sqrt{6}+\sqrt{2}-4 a 2i\sqrt{1+9\sqrt{6}-\sqrt{3}+2\sqrt{2}}.

$$x=\frac{\sqrt{2}+\sqrt{6}-4+2i\sqrt{2\sqrt{2}+9\sqrt{6}+1-\sqrt{3}}}{2\sqrt{6}+2}$$

Divide \sqrt{6}+\sqrt{2}-4+2i\sqrt{1+9\sqrt{6}-\sqrt{3}+2\sqrt{2}} entre 2+2\sqrt{6}.

$$x=\frac{\left(\sqrt{6}-1\right)\left(\sqrt{2}+\sqrt{6}-4+2i\sqrt{2\sqrt{2}+9\sqrt{6}+1-\sqrt{3}}\right)}{10}$$

Agora resolve a ecuación x=\frac{\sqrt{2}+\sqrt{6}-4±2i\sqrt{2\sqrt{2}+9\sqrt{6}+1-\sqrt{3}}}{2\sqrt{6}+2} se ± é menos. Resta 2i\sqrt{1+9\sqrt{6}-\sqrt{3}+2\sqrt{2}} de \sqrt{6}+\sqrt{2}-4.

$$x=\frac{-2i\sqrt{2\sqrt{2}+9\sqrt{6}+1-\sqrt{3}}+\sqrt{2}+\sqrt{6}-4}{2\sqrt{6}+2}$$

Divide \sqrt{6}+\sqrt{2}-4-2i\sqrt{1+9\sqrt{6}-\sqrt{3}+2\sqrt{2}} entre 2+2\sqrt{6}.

$$x=\frac{\left(\sqrt{6}-1\right)\left(-2i\sqrt{2\sqrt{2}+9\sqrt{6}+1-\sqrt{3}}+\sqrt{2}+\sqrt{6}-4\right)}{10}$$

A ecuación está resolta.

$$x=\frac{\left(\sqrt{6}-1\right)\left(\sqrt{2}+\sqrt{6}-4+2i\sqrt{2\sqrt{2}+9\sqrt{6}+1-\sqrt{3}}\right)}{10} x=\frac{\left(\sqrt{6}-1\right)\left(-2i\sqrt{2\sqrt{2}+9\sqrt{6}+1-\sqrt{3}}+\sqrt{2}+\sqrt{6}-4\right)}{10}$$

Pasos para completar o cadrado

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

$$3-x\sqrt{6}-x\sqrt{2}+x^{2}+x^{2}\sqrt{6}+2+4x+2=0$$

Suma 2 e 2 para obter 4.

$$3-x\sqrt{6}-x\sqrt{2}+x^{2}+x^{2}\sqrt{6}+4+4x=0$$

Resta 4 en ambos lados. Calquera valor restado de cero dá como resultado o valor negativo.

$$3-x\sqrt{6}-x\sqrt{2}+x^{2}+x^{2}\sqrt{6}+4x=-4$$

Resta 3 en ambos lados.

$$-x\sqrt{6}-x\sqrt{2}+x^{2}+x^{2}\sqrt{6}+4x=-4-3$$

Resta 3 de -4 para obter -7.

$$-x\sqrt{6}-x\sqrt{2}+x^{2}+x^{2}\sqrt{6}+4x=-7$$

Combina todos os termos que conteñan x.

$$\left(-\sqrt{6}-\sqrt{2}+4\right)x+\left(1+\sqrt{6}\right)x^{2}=-7$$

As ecuacións cadráticas coma esta pódense resolver completando o cadrado. Para completar o cadrado, a ecuación debe estar na forma x^{2}+bx=c.

$$\left(\sqrt{6}+1\right)x^{2}+\left(4-\sqrt{2}-\sqrt{6}\right)x=-7$$

Divide ambos lados entre 1+\sqrt{6}.

$$\frac{\left(\sqrt{6}+1\right)x^{2}+\left(4-\sqrt{2}-\sqrt{6}\right)x}{\sqrt{6}+1}=-\frac{7}{\sqrt{6}+1}$$

A división entre 1+\sqrt{6} desfai a multiplicación por 1+\sqrt{6}.

$$x^{2}+\frac{4-\sqrt{2}-\sqrt{6}}{\sqrt{6}+1}x=-\frac{7}{\sqrt{6}+1}$$

Divide -\sqrt{6}-\sqrt{2}+4 entre 1+\sqrt{6}.

$$x^{2}+\left(\frac{\sqrt{2}}{5}-\frac{2\sqrt{3}}{5}+\sqrt{6}-2\right)x=-\frac{7}{\sqrt{6}+1}$$

Divide -7 entre 1+\sqrt{6}.

$$x^{2}+\left(\frac{\sqrt{2}}{5}-\frac{2\sqrt{3}}{5}+\sqrt{6}-2\right)x=\frac{7-7\sqrt{6}}{5}$$

Divide -2+\sqrt{6}-\frac{2\sqrt{3}}{5}+\frac{\sqrt{2}}{5}, o coeficiente do termo x, entre 2 para obter -1+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}+\frac{\sqrt{2}}{10}. Despois, suma o cadrado de -1+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}+\frac{\sqrt{2}}{10} en ambos lados da ecuación. Este paso converte o lado esquerdo da ecuación nun cadrado perfecto.

$$x^{2}+\left(\frac{\sqrt{2}}{5}-\frac{2\sqrt{3}}{5}+\sqrt{6}-2\right)x+\left(\frac{\sqrt{2}}{10}+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}-1\right)^{2}=\frac{7-7\sqrt{6}}{5}+\left(\frac{\sqrt{2}}{10}+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}-1\right)^{2}$$

Eleva -1+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}+\frac{\sqrt{2}}{10} ao cadrado.

$$x^{2}+\left(\frac{\sqrt{2}}{5}-\frac{2\sqrt{3}}{5}+\sqrt{6}-2\right)x+\frac{3\sqrt{3}}{5}-\frac{4\sqrt{2}}{5}-\frac{26\sqrt{6}}{25}+\frac{66}{25}=\frac{7-7\sqrt{6}}{5}+\frac{3\sqrt{3}}{5}-\frac{4\sqrt{2}}{5}-\frac{26\sqrt{6}}{25}+\frac{66}{25}$$

Suma \frac{-7\sqrt{6}+7}{5} a \frac{66}{25}-\frac{26\sqrt{6}}{25}+\frac{3\sqrt{3}}{5}-\frac{4\sqrt{2}}{5}.

$$x^{2}+\left(\frac{\sqrt{2}}{5}-\frac{2\sqrt{3}}{5}+\sqrt{6}-2\right)x+\frac{3\sqrt{3}}{5}-\frac{4\sqrt{2}}{5}-\frac{26\sqrt{6}}{25}+\frac{66}{25}=\frac{3\sqrt{3}}{5}-\frac{4\sqrt{2}}{5}-\frac{61\sqrt{6}}{25}+\frac{101}{25}$$

Factoriza x^{2}+\left(\frac{\sqrt{2}}{5}-\frac{2\sqrt{3}}{5}+\sqrt{6}-2\right)x+\frac{3\sqrt{3}}{5}-\frac{4\sqrt{2}}{5}-\frac{26\sqrt{6}}{25}+\frac{66}{25}. En xeral, cando x^{2}+bx+c é un cadrado perfecto, sempre se pode factorizar como \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{\sqrt{2}}{10}+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}-1\right)^{2}=\frac{3\sqrt{3}}{5}-\frac{4\sqrt{2}}{5}-\frac{61\sqrt{6}}{25}+\frac{101}{25}$$

Obtén a raíz cadrada de ambos lados da ecuación.

$$\sqrt{\left(x+\frac{\sqrt{2}}{10}+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}-1\right)^{2}}=\sqrt{\frac{3\sqrt{3}}{5}-\frac{4\sqrt{2}}{5}-\frac{61\sqrt{6}}{25}+\frac{101}{25}}$$

Simplifica.

$$x+\frac{\sqrt{2}}{10}+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}-1=\frac{i\sqrt{-\left(15\sqrt{3}+101-20\sqrt{2}-61\sqrt{6}\right)}}{5} x+\frac{\sqrt{2}}{10}+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}-1=-\frac{i\sqrt{20\sqrt{2}+61\sqrt{6}-15\sqrt{3}-101}}{5}$$

Resta -1+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}+\frac{\sqrt{2}}{10} en ambos lados da ecuación.

$$x=\frac{i\sqrt{20\sqrt{2}+61\sqrt{6}-15\sqrt{3}-101}}{5}+\frac{\sqrt{3}}{5}-\frac{\sqrt{2}}{10}-\frac{\sqrt{6}}{2}+1 x=-\frac{i\sqrt{20\sqrt{2}+61\sqrt{6}-15\sqrt{3}-101}}{5}+\frac{\sqrt{3}}{5}-\frac{\sqrt{2}}{10}-\frac{\sqrt{6}}{2}+1$$