Solve (sqrt{3}-1)(sqrt{3}+1)-(sqrt{2}-1)^2

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

Evaluate

$$2\sqrt{2}-1\approx 1.828427125$$

Solution Steps

Consider \left(\sqrt{3}-1\right)\left(\sqrt{3}+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}. Square 1.

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

The square of \sqrt{3} is 3.

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

Subtract 1 from 3 to get 2.

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

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

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

The square of \sqrt{2} is 2.

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

Add 2 and 1 to get 3.

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

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

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

Subtract 3 from 2 to get -1.

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