Solve x+1/sqrt{3}+2x(x-3)/3+1/2(x+2)^2-frac{1+2sqrt{3}}{sqrt{3}}=0

$$\frac { x + 1 } { \sqrt { 3 } } + \frac { 2 x ( x - 3 ) } { 3 } + \frac { 1 } { 2 } ( x + 2 ) ^ { 2 } - \frac { 1 + 2 \sqrt { 3 } } { \sqrt { 3 } } = 0$$

Solve for x

$$x=-\frac{2\sqrt{3}}{7}\approx -0.494871659$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by 6, the least common multiple of 3,2.

$$2\times 3^{\frac{1}{2}}\left(x+1\right)+2\times 2x\left(x-3\right)+3\left(x+2\right)^{2}-2\times 3^{\frac{1}{2}}\left(1+2\sqrt{3}\right)=0$$

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

$$2\times 3^{\frac{1}{2}}x+2\times 3^{\frac{1}{2}}+2\times 2x\left(x-3\right)+3\left(x+2\right)^{2}-2\times 3^{\frac{1}{2}}\left(1+2\sqrt{3}\right)=0$$

Multiply 2 and 2 to get 4.

$$2\times 3^{\frac{1}{2}}x+2\times 3^{\frac{1}{2}}+4x\left(x-3\right)+3\left(x+2\right)^{2}-2\times 3^{\frac{1}{2}}\left(1+2\sqrt{3}\right)=0$$

Use the distributive property to multiply 4x by x-3.

$$2\times 3^{\frac{1}{2}}x+2\times 3^{\frac{1}{2}}+4x^{2}-12x+3\left(x+2\right)^{2}-2\times 3^{\frac{1}{2}}\left(1+2\sqrt{3}\right)=0$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

$$2\times 3^{\frac{1}{2}}x+2\times 3^{\frac{1}{2}}+4x^{2}-12x+3\left(x^{2}+4x+4\right)-2\times 3^{\frac{1}{2}}\left(1+2\sqrt{3}\right)=0$$

Use the distributive property to multiply 3 by x^{2}+4x+4.

$$2\times 3^{\frac{1}{2}}x+2\times 3^{\frac{1}{2}}+4x^{2}-12x+3x^{2}+12x+12-2\times 3^{\frac{1}{2}}\left(1+2\sqrt{3}\right)=0$$

Combine 4x^{2} and 3x^{2} to get 7x^{2}.

$$2\times 3^{\frac{1}{2}}x+2\times 3^{\frac{1}{2}}+7x^{2}-12x+12x+12-2\times 3^{\frac{1}{2}}\left(1+2\sqrt{3}\right)=0$$

Combine -12x and 12x to get 0.

$$2\times 3^{\frac{1}{2}}x+2\times 3^{\frac{1}{2}}+7x^{2}+12-2\times 3^{\frac{1}{2}}\left(1+2\sqrt{3}\right)=0$$

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

$$2\times 3^{\frac{1}{2}}x+2\times 3^{\frac{1}{2}}+7x^{2}+12-\left(2\times 3^{\frac{1}{2}}+4\sqrt{3}\times 3^{\frac{1}{2}}\right)=0$$

To find the opposite of 2\times 3^{\frac{1}{2}}+4\sqrt{3}\times 3^{\frac{1}{2}}, find the opposite of each term.

$$2\times 3^{\frac{1}{2}}x+2\times 3^{\frac{1}{2}}+7x^{2}+12-2\times 3^{\frac{1}{2}}-4\sqrt{3}\times 3^{\frac{1}{2}}=0$$

Combine 2\times 3^{\frac{1}{2}} and -2\times 3^{\frac{1}{2}} to get 0.

$$2\times 3^{\frac{1}{2}}x+7x^{2}+12-4\sqrt{3}\times 3^{\frac{1}{2}}=0$$

Reorder the terms.

$$7x^{2}+2\sqrt{3}x-4\sqrt{3}\sqrt{3}+12=0$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$7x^{2}+2\sqrt{3}x-4\times 3+12=0$$

Multiply -4 and 3 to get -12.

$$7x^{2}+2\sqrt{3}x-12+12=0$$

Add -12 and 12 to get 0.

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

Factor out x.

$$x\left(7x+2\sqrt{3}\right)=0$$

To find equation solutions, solve x=0 and 7x+2\sqrt{3}=0.

$$x=0 x=-\frac{2\sqrt{3}}{7}$$