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

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

Evaluate

$$9-6\sqrt{3}\approx -1.392304845$$

Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

$$12-4\sqrt{3}+1-\left(\sqrt{3}+1\right)^{2}$$

Add 12 and 1 to get 13.

$$13-4\sqrt{3}-\left(\sqrt{3}+1\right)^{2}$$

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

$$13-4\sqrt{3}-\left(\left(\sqrt{3}\right)^{2}+2\sqrt{3}+1\right)$$

The square of \sqrt{3} is 3.

$$13-4\sqrt{3}-\left(3+2\sqrt{3}+1\right)$$

Add 3 and 1 to get 4.

$$13-4\sqrt{3}-\left(4+2\sqrt{3}\right)$$

To find the opposite of 4+2\sqrt{3}, find the opposite of each term.

$$13-4\sqrt{3}-4-2\sqrt{3}$$

Subtract 4 from 13 to get 9.

$$9-4\sqrt{3}-2\sqrt{3}$$

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

$$9-6\sqrt{3}$$

Expand

$$9-6\sqrt{3}$$

Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

$$12-4\sqrt{3}+1-\left(\sqrt{3}+1\right)^{2}$$

Add 12 and 1 to get 13.

$$13-4\sqrt{3}-\left(\sqrt{3}+1\right)^{2}$$

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

$$13-4\sqrt{3}-\left(\left(\sqrt{3}\right)^{2}+2\sqrt{3}+1\right)$$

The square of \sqrt{3} is 3.

$$13-4\sqrt{3}-\left(3+2\sqrt{3}+1\right)$$

Add 3 and 1 to get 4.

$$13-4\sqrt{3}-\left(4+2\sqrt{3}\right)$$

To find the opposite of 4+2\sqrt{3}, find the opposite of each term.

$$13-4\sqrt{3}-4-2\sqrt{3}$$

Subtract 4 from 13 to get 9.

$$9-4\sqrt{3}-2\sqrt{3}$$

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

$$9-6\sqrt{3}$$