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

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

Evaluate

$$5-2\sqrt{2}\approx 2.171572875$$

Solution Steps

Expand \left(-2\sqrt{2}\right)^{2}.

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

Calculate -2 to the power of 2 and get 4.

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

The square of \sqrt{2} is 2.

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

Multiply 4 and 2 to get 8.

$$8-\left(\sqrt{2}+1\right)^{2}$$

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

$$8-\left(\left(\sqrt{2}\right)^{2}+2\sqrt{2}+1\right)$$

The square of \sqrt{2} is 2.

$$8-\left(2+2\sqrt{2}+1\right)$$

Add 2 and 1 to get 3.

$$8-\left(3+2\sqrt{2}\right)$$

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

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

Subtract 3 from 8 to get 5.

$$5-2\sqrt{2}$$

Expand

$$5-2\sqrt{2}$$

Solution Steps

Expand \left(-2\sqrt{2}\right)^{2}.

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

Calculate -2 to the power of 2 and get 4.

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

The square of \sqrt{2} is 2.

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

Multiply 4 and 2 to get 8.

$$8-\left(\sqrt{2}+1\right)^{2}$$

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

$$8-\left(\left(\sqrt{2}\right)^{2}+2\sqrt{2}+1\right)$$

The square of \sqrt{2} is 2.

$$8-\left(2+2\sqrt{2}+1\right)$$

Add 2 and 1 to get 3.

$$8-\left(3+2\sqrt{2}\right)$$

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

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

Subtract 3 from 8 to get 5.

$$5-2\sqrt{2}$$