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

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

Evaluate

$$\sqrt{3}+3\approx 4.732050808$$

Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Add 1 and 3 to get 4.

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

Use the distributive property to multiply 2 by 4+2\sqrt{3}.

$$8+4\sqrt{3}-3\left(1+\sqrt{3}\right)-2$$

Use the distributive property to multiply -3 by 1+\sqrt{3}.

$$8+4\sqrt{3}-3-3\sqrt{3}-2$$

Subtract 3 from 8 to get 5.

$$5+4\sqrt{3}-3\sqrt{3}-2$$

Combine 4\sqrt{3} and -3\sqrt{3} to get \sqrt{3}.

$$5+\sqrt{3}-2$$

Subtract 2 from 5 to get 3.

$$3+\sqrt{3}$$