Solve {left(2+sqrt{3}right)}^2-4(2+sqrt{3})+2013

$${ \left(2+ \sqrt{ 3 } \right) }^{ 2 } -4(2+ \sqrt{ 3 } )+2013$$

Evaluate

$$2012$$

Solution Steps

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

$$4+4\sqrt{3}+\left(\sqrt{3}\right)^{2}-4\left(2+\sqrt{3}\right)+2013$$

The square of \sqrt{3} is 3.

$$4+4\sqrt{3}+3-4\left(2+\sqrt{3}\right)+2013$$

Add 4 and 3 to get 7.

$$7+4\sqrt{3}-4\left(2+\sqrt{3}\right)+2013$$

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

$$7+4\sqrt{3}-8-4\sqrt{3}+2013$$

Subtract 8 from 7 to get -1.

$$-1+4\sqrt{3}-4\sqrt{3}+2013$$

Combine 4\sqrt{3} and -4\sqrt{3} to get 0.

$$-1+2013$$

Add -1 and 2013 to get 2012.

$$2012$$

Factor

$$2^{2}\times 503$$