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

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

Evaluate

$$6$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

Add 2 and 1 to get 3.

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

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

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

The square of \sqrt{2} is 2.

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

Add 1 and 2 to get 3.

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

Add 3 and 3 to get 6.

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

Combine 2\sqrt{2} and -2\sqrt{2} to get 0.

$$6$$

Factor

$$2\times 3$$