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

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

Evaluate

$$-2\sqrt{3}-8\approx -11.464101615$$

Solution Steps

Calculate 1 to the power of 2 and get 1.

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

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

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

The square of \sqrt{3} is 3.

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

Add 3 and 4 to get 7.

$$2\sqrt{3}-1-\left(7+4\sqrt{3}\right)$$

To find the opposite of 7+4\sqrt{3}, find the opposite of each term.

$$2\sqrt{3}-1-7-4\sqrt{3}$$

Subtract 7 from -1 to get -8.

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

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

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