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

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

Evaluate

$$2\left(3\sqrt{3}+5\right)\approx 20.392304845$$

Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Add 1 and 3 to get 4.

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

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

$$12+6\sqrt{3}-2\left(1+\sqrt{3}\right)+2\sqrt{3}$$

Use the distributive property to multiply -2 by 1+\sqrt{3}.

$$12+6\sqrt{3}-2-2\sqrt{3}+2\sqrt{3}$$

Subtract 2 from 12 to get 10.

$$10+6\sqrt{3}-2\sqrt{3}+2\sqrt{3}$$

Combine 6\sqrt{3} and -2\sqrt{3} to get 4\sqrt{3}.

$$10+4\sqrt{3}+2\sqrt{3}$$

Combine 4\sqrt{3} and 2\sqrt{3} to get 6\sqrt{3}.

$$10+6\sqrt{3}$$