Solve left(3+sqrt{5}right)left(3-sqrt{5}right)-{left(sqrt{5}-1right)}^2

$$\left( 3+ \sqrt{ 5 } \right) \left( 3- \sqrt{ 5 } \right) - { \left( \sqrt{ 5 } -1 \right) }^{ 2 }$$

Evaluate

$$2\left(\sqrt{5}-1\right)\approx 2.472135955$$

Solution Steps

Consider \left(3+\sqrt{5}\right)\left(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}. Square 3.

$$9-\left(\sqrt{5}\right)^{2}-\left(\sqrt{5}-1\right)^{2}$$

The square of \sqrt{5} is 5.

$$9-5-\left(\sqrt{5}-1\right)^{2}$$

Subtract 5 from 9 to get 4.

$$4-\left(\sqrt{5}-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{5}-1\right)^{2}.

$$4-\left(\left(\sqrt{5}\right)^{2}-2\sqrt{5}+1\right)$$

The square of \sqrt{5} is 5.

$$4-\left(5-2\sqrt{5}+1\right)$$

Add 5 and 1 to get 6.

$$4-\left(6-2\sqrt{5}\right)$$

To find the opposite of 6-2\sqrt{5}, find the opposite of each term.

$$4-6+2\sqrt{5}$$

Subtract 6 from 4 to get -2.

$$-2+2\sqrt{5}$$