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

$$\left. \begin{array} { l } { ( \sqrt { 2 } - 1 ) ^ { 2 } + ( 2 \sqrt { 2 } - 1 ) ^ { 2 } } \\ { + 2 \sqrt { 2 } } \end{array} \right.$$

Evaluate

$$12-4\sqrt{2}\approx 6.343145751$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

Add 2 and 1 to get 3.

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

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

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

The square of \sqrt{2} is 2.

$$3-2\sqrt{2}+4\times 2-4\sqrt{2}+1+2\sqrt{2}$$

Multiply 4 and 2 to get 8.

$$3-2\sqrt{2}+8-4\sqrt{2}+1+2\sqrt{2}$$

Add 8 and 1 to get 9.

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

Add 3 and 9 to get 12.

$$12-2\sqrt{2}-4\sqrt{2}+2\sqrt{2}$$

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

$$12-6\sqrt{2}+2\sqrt{2}$$

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

$$12-4\sqrt{2}$$