Solve (1-sqrt{3})-(1-sqrt{3})^2-2

$$( 1 - \sqrt { 3 } ) - ( 1 - \sqrt { 3 } ) ^ { 2 } - 2$$

Evaluate

$$\sqrt{3}-5\approx -3.267949192$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\sqrt{3}\right)^{2}.

$$1-\sqrt{3}-\left(1-2\sqrt{3}+\left(\sqrt{3}\right)^{2}\right)-2$$

The square of \sqrt{3} is 3.

$$1-\sqrt{3}-\left(1-2\sqrt{3}+3\right)-2$$

Add 1 and 3 to get 4.

$$1-\sqrt{3}-\left(4-2\sqrt{3}\right)-2$$

To find the opposite of 4-2\sqrt{3}, find the opposite of each term.

$$1-\sqrt{3}-4+2\sqrt{3}-2$$

Subtract 4 from 1 to get -3.

$$-3-\sqrt{3}+2\sqrt{3}-2$$

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

$$-3+\sqrt{3}-2$$

Subtract 2 from -3 to get -5.

$$-5+\sqrt{3}$$