Solve -frac{sqrt{3}}{4}x+b=frac{sqrt{3}}{3}x^2+2/3sqrt{3}x-sqrt{3}

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

Solve for b

$$b=-\frac{\sqrt{3}\left(12-11x-4x^{2}\right)}{12}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 12, the least common multiple of 4,3.

$$12\left(-\frac{\sqrt{3}}{4}\right)x+12b=4\sqrt{3}x^{2}+8\sqrt{3}x-12\sqrt{3}$$

Cancel out 4, the greatest common factor in 12 and 4.

$$-3\sqrt{3}x+12b=4\sqrt{3}x^{2}+8\sqrt{3}x-12\sqrt{3}$$

Add 3\sqrt{3}x to both sides.

$$12b=4\sqrt{3}x^{2}+8\sqrt{3}x-12\sqrt{3}+3\sqrt{3}x$$

Combine 8\sqrt{3}x and 3\sqrt{3}x to get 11\sqrt{3}x.

$$12b=4\sqrt{3}x^{2}+11\sqrt{3}x-12\sqrt{3}$$

Divide both sides by 12.

$$\frac{12b}{12}=\frac{\sqrt{3}\left(4x^{2}+11x-12\right)}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$b=\frac{\sqrt{3}\left(4x^{2}+11x-12\right)}{12}$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{64\sqrt{3}b+313}-11}{8}$$

$$x=\frac{\sqrt{64\sqrt{3}b+313}-11}{8}$$

Solve for x

$$x=\frac{-\sqrt{64\sqrt{3}b+313}-11}{8}$$

$$x=\frac{\sqrt{64\sqrt{3}b+313}-11}{8}\text{, }b\geq -\frac{313\sqrt{3}}{192}$$