Solve a^2+a^2/2+2a^2/2-sqrt{3}=

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

Factor

$$\frac{\left(4\sqrt{3}+11\right)a^{2}}{2}$$

Solution Steps

Combine a^{2} and \frac{a^{2}}{2} to get \frac{3}{2}a^{2}.

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

Rationalize the denominator of \frac{2a^{2}}{2-\sqrt{3}} by multiplying numerator and denominator by 2+\sqrt{3}.

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

Consider \left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square 2. Square \sqrt{3}.

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

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

$$factor(\frac{3}{2}a^{2}+2a^{2}\left(2+\sqrt{3}\right))$$

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

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

Combine \frac{3}{2}a^{2} and 4a^{2} to get \frac{11}{2}a^{2}.

$$factor(\frac{11}{2}a^{2}+2a^{2}\sqrt{3})$$

Factor out \frac{1}{2}.

$$\frac{11a^{2}+4a^{2}\sqrt{3}}{2}$$

Consider 11a^{2}+4a^{2}\sqrt{3}. Factor out a^{2}.

$$a^{2}\left(11+4\sqrt{3}\right)$$

Rewrite the complete factored expression.

$$\frac{a^{2}\left(11+4\sqrt{3}\right)}{2}$$

Evaluate

$$\frac{\left(4\sqrt{3}+11\right)a^{2}}{2}$$