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

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

Solve for x (complex solution)

$$x=\frac{\left(\sqrt{6}-1\right)\left(\sqrt{2}+\sqrt{6}-4+2i\sqrt{2\sqrt{2}+12\sqrt{6}+4-\sqrt{3}}\right)}{10}\approx -0.019756066+1.702524356i$$

$$x=\frac{\left(\sqrt{6}-1\right)\left(-2i\sqrt{2\sqrt{2}+12\sqrt{6}+4-\sqrt{3}}+\sqrt{2}+\sqrt{6}-4\right)}{10}\approx -0.019756066-1.702524356i$$

Steps Using the Quadratic Formula

Combine -x^{2} and 2x^{2} to get x^{2}.

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

Add 2 and 3 to get 5.

$$3-x\sqrt{6}-x\sqrt{2}+x^{2}+x^{2}\sqrt{6}+5+4x+2=0$$

Add 5 and 2 to get 7.

$$3-x\sqrt{6}-x\sqrt{2}+x^{2}+x^{2}\sqrt{6}+7+4x=0$$

Add 3 and 7 to get 10.

$$10-x\sqrt{6}-x\sqrt{2}+x^{2}+x^{2}\sqrt{6}+4x=0$$

Combine all terms containing x.

$$10+\left(-\sqrt{6}-\sqrt{2}+4\right)x+\left(1+\sqrt{6}\right)x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$\left(\sqrt{6}+1\right)x^{2}+\left(4-\sqrt{2}-\sqrt{6}\right)x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+\sqrt{6} for a, -\sqrt{6}-\sqrt{2}+4 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(4-\sqrt{2}-\sqrt{6}\right)±\sqrt{\left(4-\sqrt{2}-\sqrt{6}\right)^{2}-4\left(\sqrt{6}+1\right)\times 10}}{2\left(\sqrt{6}+1\right)}$$

Square -\sqrt{6}-\sqrt{2}+4.

$$x=\frac{-\left(4-\sqrt{2}-\sqrt{6}\right)±\sqrt{4\sqrt{3}+24-8\sqrt{2}-8\sqrt{6}-4\left(\sqrt{6}+1\right)\times 10}}{2\left(\sqrt{6}+1\right)}$$

Multiply -4 times 1+\sqrt{6}.

$$x=\frac{-\left(4-\sqrt{2}-\sqrt{6}\right)±\sqrt{4\sqrt{3}+24-8\sqrt{2}-8\sqrt{6}+\left(-4\sqrt{6}-4\right)\times 10}}{2\left(\sqrt{6}+1\right)}$$

Multiply -4-4\sqrt{6} times 10.

$$x=\frac{-\left(4-\sqrt{2}-\sqrt{6}\right)±\sqrt{4\sqrt{3}+24-8\sqrt{2}-8\sqrt{6}-40\sqrt{6}-40}}{2\left(\sqrt{6}+1\right)}$$

Add 24-8\sqrt{6}+4\sqrt{3}-8\sqrt{2} to -40-40\sqrt{6}.

$$x=\frac{-\left(4-\sqrt{2}-\sqrt{6}\right)±\sqrt{4\sqrt{3}-8\sqrt{2}-48\sqrt{6}-16}}{2\left(\sqrt{6}+1\right)}$$

Take the square root of -16-48\sqrt{6}+4\sqrt{3}-8\sqrt{2}.

$$x=\frac{-\left(4-\sqrt{2}-\sqrt{6}\right)±2i\sqrt{2\sqrt{2}+12\sqrt{6}+4-\sqrt{3}}}{2\left(\sqrt{6}+1\right)}$$

Multiply 2 times 1+\sqrt{6}.

$$x=\frac{\sqrt{2}+\sqrt{6}-4±2i\sqrt{2\sqrt{2}+12\sqrt{6}+4-\sqrt{3}}}{2\sqrt{6}+2}$$

Now solve the equation x=\frac{\sqrt{2}+\sqrt{6}-4±2i\sqrt{2\sqrt{2}+12\sqrt{6}+4-\sqrt{3}}}{2\sqrt{6}+2} when ± is plus. Add \sqrt{6}+\sqrt{2}-4 to 2i\sqrt{4+12\sqrt{6}-\sqrt{3}+2\sqrt{2}}.

$$x=\frac{\sqrt{2}+\sqrt{6}-4+2i\sqrt{2\sqrt{2}+12\sqrt{6}+4-\sqrt{3}}}{2\sqrt{6}+2}$$

Divide \sqrt{6}+\sqrt{2}-4+2i\sqrt{4+12\sqrt{6}-\sqrt{3}+2\sqrt{2}} by 2+2\sqrt{6}.

$$x=\frac{\left(\sqrt{6}-1\right)\left(\sqrt{2}+\sqrt{6}-4+2i\sqrt{2\sqrt{2}+12\sqrt{6}+4-\sqrt{3}}\right)}{10}$$

Now solve the equation x=\frac{\sqrt{2}+\sqrt{6}-4±2i\sqrt{2\sqrt{2}+12\sqrt{6}+4-\sqrt{3}}}{2\sqrt{6}+2} when ± is minus. Subtract 2i\sqrt{4+12\sqrt{6}-\sqrt{3}+2\sqrt{2}} from \sqrt{6}+\sqrt{2}-4.

$$x=\frac{-2i\sqrt{2\sqrt{2}+12\sqrt{6}+4-\sqrt{3}}+\sqrt{2}+\sqrt{6}-4}{2\sqrt{6}+2}$$

Divide \sqrt{6}+\sqrt{2}-4-2i\sqrt{4+12\sqrt{6}-\sqrt{3}+2\sqrt{2}} by 2+2\sqrt{6}.

$$x=\frac{\left(\sqrt{6}-1\right)\left(-2i\sqrt{2\sqrt{2}+12\sqrt{6}+4-\sqrt{3}}+\sqrt{2}+\sqrt{6}-4\right)}{10}$$

The equation is now solved.

$$x=\frac{\left(\sqrt{6}-1\right)\left(\sqrt{2}+\sqrt{6}-4+2i\sqrt{2\sqrt{2}+12\sqrt{6}+4-\sqrt{3}}\right)}{10} x=\frac{\left(\sqrt{6}-1\right)\left(-2i\sqrt{2\sqrt{2}+12\sqrt{6}+4-\sqrt{3}}+\sqrt{2}+\sqrt{6}-4\right)}{10}$$

Steps for Completing the Square

Combine -x^{2} and 2x^{2} to get x^{2}.

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

Add 2 and 3 to get 5.

$$3-x\sqrt{6}-x\sqrt{2}+x^{2}+x^{2}\sqrt{6}+5+4x+2=0$$

Add 5 and 2 to get 7.

$$3-x\sqrt{6}-x\sqrt{2}+x^{2}+x^{2}\sqrt{6}+7+4x=0$$

Subtract 7 from both sides. Anything subtracted from zero gives its negation.

$$3-x\sqrt{6}-x\sqrt{2}+x^{2}+x^{2}\sqrt{6}+4x=-7$$

Subtract 3 from both sides.

$$-x\sqrt{6}-x\sqrt{2}+x^{2}+x^{2}\sqrt{6}+4x=-7-3$$

Subtract 3 from -7 to get -10.

$$-x\sqrt{6}-x\sqrt{2}+x^{2}+x^{2}\sqrt{6}+4x=-10$$

Combine all terms containing x.

$$\left(-\sqrt{6}-\sqrt{2}+4\right)x+\left(1+\sqrt{6}\right)x^{2}=-10$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$\left(\sqrt{6}+1\right)x^{2}+\left(4-\sqrt{2}-\sqrt{6}\right)x=-10$$

Divide both sides by 1+\sqrt{6}.

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

Dividing by 1+\sqrt{6} undoes the multiplication by 1+\sqrt{6}.

$$x^{2}+\frac{4-\sqrt{2}-\sqrt{6}}{\sqrt{6}+1}x=-\frac{10}{\sqrt{6}+1}$$

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

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

Divide -10 by 1+\sqrt{6}.

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

Divide -2+\sqrt{6}-\frac{2\sqrt{3}}{5}+\frac{\sqrt{2}}{5}, the coefficient of the x term, by 2 to get -1+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}+\frac{\sqrt{2}}{10}. Then add the square of -1+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}+\frac{\sqrt{2}}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -1+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}+\frac{\sqrt{2}}{10}.

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

Add -2\sqrt{6}+2 to \frac{66}{25}-\frac{26\sqrt{6}}{25}+\frac{3\sqrt{3}}{5}-\frac{4\sqrt{2}}{5}.

$$x^{2}+\left(\frac{\sqrt{2}}{5}-\frac{2\sqrt{3}}{5}+\sqrt{6}-2\right)x+\frac{3\sqrt{3}}{5}-\frac{4\sqrt{2}}{5}-\frac{26\sqrt{6}}{25}+\frac{66}{25}=\frac{3\sqrt{3}}{5}-\frac{4\sqrt{2}}{5}-\frac{76\sqrt{6}}{25}+\frac{116}{25}$$

Factor 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}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{\sqrt{2}}{10}+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}-1\right)^{2}=\frac{3\sqrt{3}}{5}-\frac{4\sqrt{2}}{5}-\frac{76\sqrt{6}}{25}+\frac{116}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{\sqrt{2}}{10}+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}-1\right)^{2}}=\sqrt{\frac{3\sqrt{3}}{5}-\frac{4\sqrt{2}}{5}-\frac{76\sqrt{6}}{25}+\frac{116}{25}}$$

Simplify.

$$x+\frac{\sqrt{2}}{10}+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}-1=\frac{i\sqrt{-\left(15\sqrt{3}+116-20\sqrt{2}-76\sqrt{6}\right)}}{5} x+\frac{\sqrt{2}}{10}+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}-1=-\frac{i\sqrt{20\sqrt{2}+76\sqrt{6}-15\sqrt{3}-116}}{5}$$

Subtract -1+\frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{5}+\frac{\sqrt{2}}{10} from both sides of the equation.

$$x=\frac{i\sqrt{20\sqrt{2}+76\sqrt{6}-15\sqrt{3}-116}}{5}+\frac{\sqrt{3}}{5}-\frac{\sqrt{2}}{10}-\frac{\sqrt{6}}{2}+1 x=-\frac{i\sqrt{20\sqrt{2}+76\sqrt{6}-15\sqrt{3}-116}}{5}+\frac{\sqrt{3}}{5}-\frac{\sqrt{2}}{10}-\frac{\sqrt{6}}{2}+1$$