Solve a=(1/2-sqrt{3})+(frac{sqrt{3}}{2}+1)+(1-sqrt{5})*(1+sqrt{5})

$$a = ( \frac { 1 } { 2 } - \sqrt { 3 } ) + ( \frac { \sqrt { 3 } } { 2 } + 1 ) + ( 1 - \sqrt { 5 } ) \cdot ( 1 + \sqrt { 5 } )$$

Solve for a

$$a=\frac{-\sqrt{3}-5}{2}\approx -3.366025404$$

Steps for Solving Linear Equation

Combine -\sqrt{3} and \frac{\sqrt{3}}{2} to get -\frac{1}{2}\sqrt{3}.

$$a=\frac{1}{2}-\frac{1}{2}\sqrt{3}+1+\left(1-\sqrt{5}\right)\left(1+\sqrt{5}\right)$$

Convert 1 to fraction \frac{2}{2}.

$$a=\frac{1}{2}-\frac{1}{2}\sqrt{3}+\frac{2}{2}+\left(1-\sqrt{5}\right)\left(1+\sqrt{5}\right)$$

Since \frac{1}{2} and \frac{2}{2} have the same denominator, add them by adding their numerators.

$$a=\frac{1+2}{2}-\frac{1}{2}\sqrt{3}+\left(1-\sqrt{5}\right)\left(1+\sqrt{5}\right)$$

Add 1 and 2 to get 3.

$$a=\frac{3}{2}-\frac{1}{2}\sqrt{3}+\left(1-\sqrt{5}\right)\left(1+\sqrt{5}\right)$$

Consider \left(1-\sqrt{5}\right)\left(1+\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$a=\frac{3}{2}-\frac{1}{2}\sqrt{3}+1^{2}-\left(\sqrt{5}\right)^{2}$$

Calculate 1 to the power of 2 and get 1.

$$a=\frac{3}{2}-\frac{1}{2}\sqrt{3}+1-\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$a=\frac{3}{2}-\frac{1}{2}\sqrt{3}+1-5$$

Subtract 5 from 1 to get -4.

$$a=\frac{3}{2}-\frac{1}{2}\sqrt{3}-4$$

Convert 4 to fraction \frac{8}{2}.

$$a=\frac{3}{2}-\frac{1}{2}\sqrt{3}-\frac{8}{2}$$

Since \frac{3}{2} and \frac{8}{2} have the same denominator, subtract them by subtracting their numerators.

$$a=\frac{3-8}{2}-\frac{1}{2}\sqrt{3}$$

Subtract 8 from 3 to get -5.

$$a=-\frac{5}{2}-\frac{1}{2}\sqrt{3}$$

Assign a

$$a≔\frac{-\sqrt{3}-5}{2}$$