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

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

Solve for x (complex solution)

$$x=\frac{\sqrt{6}i}{2}+\frac{2\sqrt{3}}{4}+\frac{3\sqrt{2}}{6}-\frac{1}{2}i\approx 1.573132185+0.724744871i$$

$$x=-\frac{\sqrt{6}i}{2}+\frac{2\sqrt{3}}{4}+\frac{3\sqrt{2}}{6}+\frac{1}{2}i\approx 1.573132185-0.724744871i$$

Steps Using the Quadratic Formula

Combine all terms containing x.

$$\sqrt{6}x^{2}+\left(-2\sqrt{3}-3\sqrt{2}\right)x+3\sqrt{6}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \sqrt{6} for a, -2\sqrt{3}-3\sqrt{2} for b, and 3\sqrt{6} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\sqrt{3}-3\sqrt{2}\right)±\sqrt{\left(-2\sqrt{3}-3\sqrt{2}\right)^{2}-4\sqrt{6}\times 3\sqrt{6}}}{2\sqrt{6}}$$

Square -2\sqrt{3}-3\sqrt{2}.

$$x=\frac{-\left(-2\sqrt{3}-3\sqrt{2}\right)±\sqrt{12\sqrt{6}+30-4\sqrt{6}\times 3\sqrt{6}}}{2\sqrt{6}}$$

Multiply -4 times \sqrt{6}.

$$x=\frac{-\left(-2\sqrt{3}-3\sqrt{2}\right)±\sqrt{12\sqrt{6}+30+\left(-4\sqrt{6}\right)\times 3\sqrt{6}}}{2\sqrt{6}}$$

Multiply -4\sqrt{6} times 3\sqrt{6}.

$$x=\frac{-\left(-2\sqrt{3}-3\sqrt{2}\right)±\sqrt{12\sqrt{6}+30-72}}{2\sqrt{6}}$$

Add 30+12\sqrt{6} to -72.

$$x=\frac{-\left(-2\sqrt{3}-3\sqrt{2}\right)±\sqrt{12\sqrt{6}-42}}{2\sqrt{6}}$$

Take the square root of -42+12\sqrt{6}.

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

The opposite of -2\sqrt{3}-3\sqrt{2} is 2\sqrt{3}+3\sqrt{2}.

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

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

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

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

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

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

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

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

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

The equation is now solved.

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

Steps for Completing the Square

Subtract 3\sqrt{6} from both sides. Anything subtracted from zero gives its negation.

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

Combine all terms containing x.

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

Divide both sides by \sqrt{6}.

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

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

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

Divide -2\sqrt{3}-3\sqrt{2} by \sqrt{6}.

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

Divide -3\sqrt{6} by \sqrt{6}.

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

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

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

Square \frac{-\sqrt{2}-\sqrt{3}}{2}.

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

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

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

Factor x^{2}+\left(-\sqrt{2}-\sqrt{3}\right)x+\frac{\sqrt{6}}{2}+\frac{5}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Subtract \frac{-\sqrt{2}-\sqrt{3}}{2} from both sides of the equation.

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