Solve (4-2sqrt{6})^2-(1+sqrt{6})^2=

$$( 4 - 2 \sqrt { 6 } ) ^ { 2 } - ( 1 + \sqrt { 6 } ) ^ { 2 } =$$

Evaluate

$$33-18\sqrt{6}\approx -11.09081537$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-2\sqrt{6}\right)^{2}.

$$16-16\sqrt{6}+4\left(\sqrt{6}\right)^{2}-\left(1+\sqrt{6}\right)^{2}$$

The square of \sqrt{6} is 6.

$$16-16\sqrt{6}+4\times 6-\left(1+\sqrt{6}\right)^{2}$$

Multiply 4 and 6 to get 24.

$$16-16\sqrt{6}+24-\left(1+\sqrt{6}\right)^{2}$$

Add 16 and 24 to get 40.

$$40-16\sqrt{6}-\left(1+\sqrt{6}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\sqrt{6}\right)^{2}.

$$40-16\sqrt{6}-\left(1+2\sqrt{6}+\left(\sqrt{6}\right)^{2}\right)$$

The square of \sqrt{6} is 6.

$$40-16\sqrt{6}-\left(1+2\sqrt{6}+6\right)$$

Add 1 and 6 to get 7.

$$40-16\sqrt{6}-\left(7+2\sqrt{6}\right)$$

To find the opposite of 7+2\sqrt{6}, find the opposite of each term.

$$40-16\sqrt{6}-7-2\sqrt{6}$$

Subtract 7 from 40 to get 33.

$$33-16\sqrt{6}-2\sqrt{6}$$

Combine -16\sqrt{6} and -2\sqrt{6} to get -18\sqrt{6}.

$$33-18\sqrt{6}$$

Expand

$$33-18\sqrt{6}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-2\sqrt{6}\right)^{2}.

$$16-16\sqrt{6}+4\left(\sqrt{6}\right)^{2}-\left(1+\sqrt{6}\right)^{2}$$

The square of \sqrt{6} is 6.

$$16-16\sqrt{6}+4\times 6-\left(1+\sqrt{6}\right)^{2}$$

Multiply 4 and 6 to get 24.

$$16-16\sqrt{6}+24-\left(1+\sqrt{6}\right)^{2}$$

Add 16 and 24 to get 40.

$$40-16\sqrt{6}-\left(1+\sqrt{6}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\sqrt{6}\right)^{2}.

$$40-16\sqrt{6}-\left(1+2\sqrt{6}+\left(\sqrt{6}\right)^{2}\right)$$

The square of \sqrt{6} is 6.

$$40-16\sqrt{6}-\left(1+2\sqrt{6}+6\right)$$

Add 1 and 6 to get 7.

$$40-16\sqrt{6}-\left(7+2\sqrt{6}\right)$$

To find the opposite of 7+2\sqrt{6}, find the opposite of each term.

$$40-16\sqrt{6}-7-2\sqrt{6}$$

Subtract 7 from 40 to get 33.

$$33-16\sqrt{6}-2\sqrt{6}$$

Combine -16\sqrt{6} and -2\sqrt{6} to get -18\sqrt{6}.

$$33-18\sqrt{6}$$