Solve 4=a^2+(-frac{sqrt{3}}{3}a+4-2)^2

$$4 = a ^ { 2 } + ( - \frac { \sqrt { 3 } } { 3 } a + 4 - 2 ) ^ { 2 }$$

Solve for a

$$a=\sqrt{3}\approx 1.732050808$$

$$a=0$$

Steps Using Factoring

Express \left(-\frac{\sqrt{3}}{3}\right)a as a single fraction.

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

Subtract 2 from 4 to get 2.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{-\sqrt{3}a}{3}+2\right)^{2}.

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

To raise \frac{-\sqrt{3}a}{3} to a power, raise both numerator and denominator to the power and then divide.

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

Express 4\times \frac{-\sqrt{3}a}{3} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3^{2} and 3 is 9. Multiply \frac{4\left(-1\right)\sqrt{3}a}{3} times \frac{3}{3}.

$$4=a^{2}+\frac{\left(-\sqrt{3}a\right)^{2}}{9}+\frac{3\times 4\left(-1\right)\sqrt{3}a}{9}+4$$

Since \frac{\left(-\sqrt{3}a\right)^{2}}{9} and \frac{3\times 4\left(-1\right)\sqrt{3}a}{9} have the same denominator, add them by adding their numerators.

$$4=a^{2}+\frac{\left(-\sqrt{3}a\right)^{2}+3\times 4\left(-1\right)\sqrt{3}a}{9}+4$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{3^{2}}{3^{2}}.

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

Since \frac{\left(-\sqrt{3}a\right)^{2}}{3^{2}} and \frac{4\times 3^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

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

Multiply 4 and -1 to get -4.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply a^{2} times \frac{3^{2}}{3^{2}}.

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

Since \frac{a^{2}\times 3^{2}}{3^{2}} and \frac{\left(-\sqrt{3}a\right)^{2}+4\times 3^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3^{2} and 3 is 9. Multiply \frac{-4\sqrt{3}a}{3} times \frac{3}{3}.

$$4=a^{2}+\frac{\left(-\sqrt{3}a\right)^{2}+4\times 3^{2}}{9}+\frac{3\left(-4\right)\sqrt{3}a}{9}$$

Since \frac{\left(-\sqrt{3}a\right)^{2}+4\times 3^{2}}{9} and \frac{3\left(-4\right)\sqrt{3}a}{9} have the same denominator, add them by adding their numerators.

$$4=a^{2}+\frac{\left(-\sqrt{3}a\right)^{2}+4\times 3^{2}+3\left(-4\right)\sqrt{3}a}{9}$$

Expand \left(-\sqrt{3}a\right)^{2}.

$$4=a^{2}+\frac{\left(-1\right)^{2}\left(\sqrt{3}\right)^{2}a^{2}+4\times 3^{2}+3\left(-4\right)\sqrt{3}a}{9}$$

Calculate -1 to the power of 2 and get 1.

$$4=a^{2}+\frac{1\left(\sqrt{3}\right)^{2}a^{2}+4\times 3^{2}+3\left(-4\right)\sqrt{3}a}{9}$$

The square of \sqrt{3} is 3.

$$4=a^{2}+\frac{1\times 3a^{2}+4\times 3^{2}+3\left(-4\right)\sqrt{3}a}{9}$$

Multiply 1 and 3 to get 3.

$$4=a^{2}+\frac{3a^{2}+4\times 3^{2}+3\left(-4\right)\sqrt{3}a}{9}$$

Calculate 3 to the power of 2 and get 9.

$$4=a^{2}+\frac{3a^{2}+4\times 9+3\left(-4\right)\sqrt{3}a}{9}$$

Multiply 4 and 9 to get 36.

$$4=a^{2}+\frac{3a^{2}+36+3\left(-4\right)\sqrt{3}a}{9}$$

Multiply 3 and -4 to get -12.

$$4=a^{2}+\frac{3a^{2}+36-12\sqrt{3}a}{9}$$

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

$$a^{2}+\frac{3a^{2}+36-12\sqrt{3}a}{9}=4$$

Subtract 4 from both sides.

$$a^{2}+\frac{3a^{2}+36-12\sqrt{3}a}{9}-4=0$$

Multiply both sides of the equation by 9.

$$9a^{2}+3a^{2}+36-12\sqrt{3}a-36=0$$

Reorder the terms.

$$3a^{2}+9a^{2}-12\sqrt{3}a+36-36=0$$

Combine 3a^{2} and 9a^{2} to get 12a^{2}.

$$12a^{2}-12\sqrt{3}a+36-36=0$$

Subtract 36 from 36 to get 0.

$$12a^{2}-12\sqrt{3}a=0$$

Factor out a.

$$a\left(12a-12\sqrt{3}\right)=0$$

To find equation solutions, solve a=0 and 12a-12\sqrt{3}=0.

$$a=0 a=\sqrt{3}$$