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

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

Evaluate

$$-8\sqrt{3}-1\approx -14.856406461$$

Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Add 4 and 3 to get 7.

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

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

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

Subtract 8 from 7 to get -1.

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

Combine -4\sqrt{3} and -4\sqrt{3} to get -8\sqrt{3}.

$$-1-8\sqrt{3}$$