Solve frac{sqrt{2}}{2}2+1/sqrt{3}=3x^2+15/sqrt{3}

$$\frac{ \sqrt{ 2 } }{ 2 } 2+ \frac{ 1 }{ \sqrt{ 3 } } = \frac{ 3 { x }^{ 2 } +15 }{ \sqrt{ 3 } }$$

Solve for x (complex solution)

$$x=-\frac{i\sqrt{42-3\sqrt{6}}}{3}\approx -0-1.962185028i$$

$$x=\frac{i\sqrt{42-3\sqrt{6}}}{3}\approx 1.962185028i$$

Steps by Finding Square Root

Multiply both sides of the equation by 2.

$$2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}=\frac{2}{3}\times 3^{\frac{1}{2}}\left(3x^{2}+15\right)$$

Use the distributive property to multiply \frac{2}{3}\times 3^{\frac{1}{2}} by 3x^{2}+15.

$$2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}=2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}$$

Swap sides so that all variable terms are on the left hand side.

$$2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}=2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}$$

Subtract 10\times 3^{\frac{1}{2}} from both sides.

$$2\times 3^{\frac{1}{2}}x^{2}=2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}-10\times 3^{\frac{1}{2}}$$

Combine \frac{2}{3}\times 3^{\frac{1}{2}} and -10\times 3^{\frac{1}{2}} to get -\frac{28}{3}\times 3^{\frac{1}{2}}.

$$2\times 3^{\frac{1}{2}}x^{2}=2\sqrt{2}-\frac{28}{3}\times 3^{\frac{1}{2}}$$

Reorder the terms.

$$2\sqrt{3}x^{2}=-\frac{28}{3}\sqrt{3}+2\sqrt{2}$$

Dividing by 2\sqrt{3} undoes the multiplication by 2\sqrt{3}.

$$x^{2}=\frac{-\frac{28\sqrt{3}}{3}+2\sqrt{2}}{2\sqrt{3}}$$

Divide -\frac{28\sqrt{3}}{3}+2\sqrt{2} by 2\sqrt{3}.

$$x^{2}=\frac{\sqrt{6}-14}{3}$$

Take the square root of both sides of the equation.

$$x=\frac{i\sqrt{42-3\sqrt{6}}}{3} x=-\frac{i\sqrt{42-3\sqrt{6}}}{3}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}=\frac{2}{3}\times 3^{\frac{1}{2}}\left(3x^{2}+15\right)$$

Use the distributive property to multiply \frac{2}{3}\times 3^{\frac{1}{2}} by 3x^{2}+15.

$$2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}=2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}$$

Swap sides so that all variable terms are on the left hand side.

$$2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}=2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}$$

Subtract 2\sqrt{2} from both sides.

$$2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}-2\sqrt{2}=\frac{2}{3}\times 3^{\frac{1}{2}}$$

Subtract \frac{2}{3}\times 3^{\frac{1}{2}} from both sides.

$$2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}-2\sqrt{2}-\frac{2}{3}\times 3^{\frac{1}{2}}=0$$

Combine 10\times 3^{\frac{1}{2}} and -\frac{2}{3}\times 3^{\frac{1}{2}} to get \frac{28}{3}\times 3^{\frac{1}{2}}.

$$2\times 3^{\frac{1}{2}}x^{2}+\frac{28}{3}\times 3^{\frac{1}{2}}-2\sqrt{2}=0$$

Reorder the terms.

$$2\sqrt{3}x^{2}-2\sqrt{2}+\frac{28}{3}\sqrt{3}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$2\sqrt{3}x^{2}+\frac{28\sqrt{3}}{3}-2\sqrt{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2\sqrt{3} for a, 0 for b, and -2\sqrt{2}+\frac{28\sqrt{3}}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 2\sqrt{3}\left(\frac{28\sqrt{3}}{3}-2\sqrt{2}\right)}}{2\times 2\sqrt{3}}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 2\sqrt{3}\left(\frac{28\sqrt{3}}{3}-2\sqrt{2}\right)}}{2\times 2\sqrt{3}}$$

Multiply -4 times 2\sqrt{3}.

$$x=\frac{0±\sqrt{\left(-8\sqrt{3}\right)\left(\frac{28\sqrt{3}}{3}-2\sqrt{2}\right)}}{2\times 2\sqrt{3}}$$

Multiply -8\sqrt{3} times -2\sqrt{2}+\frac{28\sqrt{3}}{3}.

$$x=\frac{0±\sqrt{16\sqrt{6}-224}}{2\times 2\sqrt{3}}$$

Take the square root of 16\sqrt{6}-224.

$$x=\frac{0±4i\sqrt{14-\sqrt{6}}}{2\times 2\sqrt{3}}$$

Multiply 2 times 2\sqrt{3}.

$$x=\frac{0±4i\sqrt{14-\sqrt{6}}}{4\sqrt{3}}$$

Now solve the equation x=\frac{0±4i\sqrt{14-\sqrt{6}}}{4\sqrt{3}} when ± is plus.

$$x=\frac{i\sqrt{42-3\sqrt{6}}}{3}$$

Now solve the equation x=\frac{0±4i\sqrt{14-\sqrt{6}}}{4\sqrt{3}} when ± is minus.

$$x=-\frac{i\sqrt{42-3\sqrt{6}}}{3}$$

The equation is now solved.

$$x=\frac{i\sqrt{42-3\sqrt{6}}}{3} x=-\frac{i\sqrt{42-3\sqrt{6}}}{3}$$