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

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

Evaluate

$$8-\sqrt{2}\approx 6.585786438$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

$$\sqrt{2}-1+4\times 2-4\sqrt{2}+1+2\sqrt{2}$$

Multiply 4 and 2 to get 8.

$$\sqrt{2}-1+8-4\sqrt{2}+1+2\sqrt{2}$$

Add 8 and 1 to get 9.

$$\sqrt{2}-1+9-4\sqrt{2}+2\sqrt{2}$$

Add -1 and 9 to get 8.

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

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

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

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

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