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

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

Evaluate

$$2-3\sqrt{2}\approx -2.242640687$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

Add 1 and 2 to get 3.

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

Rationalize the denominator of \frac{1}{\sqrt{2}-1} by multiplying numerator and denominator by \sqrt{2}+1.

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

Consider \left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{2}. Square 1.

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

Subtract 1 from 2 to get 1.

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

Anything divided by one gives itself.

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

To find the opposite of \sqrt{2}+1, find the opposite of each term.

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

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

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

Subtract 1 from 3 to get 2.

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