Solve (sqrt{3}-sqrt{5})(sqrt{3}+sqrt{5})-(sqrt{3}-sqrt{5})^2

$$( \sqrt { 3 } - \sqrt { 5 } ) ( \sqrt { 3 } + \sqrt { 5 } ) - ( \sqrt { 3 } - \sqrt { 5 } ) ^ { 2 }$$

Evaluate

$$2\left(\sqrt{15}-5\right)\approx -2.254033308$$

Solution Steps

Consider \left(\sqrt{3}-\sqrt{5}\right)\left(\sqrt{3}+\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(\sqrt{3}\right)^{2}-\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}-\sqrt{5}\right)^{2}$$

The square of \sqrt{3} is 3.

$$3-\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}-\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$3-5-\left(\sqrt{3}-\sqrt{5}\right)^{2}$$

Subtract 5 from 3 to get -2.

$$-2-\left(\sqrt{3}-\sqrt{5}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{3}-\sqrt{5}\right)^{2}.

$$-2-\left(\left(\sqrt{3}\right)^{2}-2\sqrt{3}\sqrt{5}+\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{3} is 3.

$$-2-\left(3-2\sqrt{3}\sqrt{5}+\left(\sqrt{5}\right)^{2}\right)$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$-2-\left(3-2\sqrt{15}+\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$-2-\left(3-2\sqrt{15}+5\right)$$

Add 3 and 5 to get 8.

$$-2-\left(8-2\sqrt{15}\right)$$

To find the opposite of 8-2\sqrt{15}, find the opposite of each term.

$$-2-8+2\sqrt{15}$$

Subtract 8 from -2 to get -10.

$$-10+2\sqrt{15}$$

Factor

$$2 {(\sqrt{15} - 5)} = -2.254033308$$