Solve (4)(3+sqrt{5})(3-sqrt{5})-(sqrt{5}-1)^2

$$( 4 ) ( 3 + \sqrt { 5 } ) ( 3 - \sqrt { 5 } ) - ( \sqrt { 5 } - 1 ) ^ { 2 }$$

Evaluate

$$2\sqrt{5}+10\approx 14.472135955$$

Solution Steps

Use the distributive property to multiply 4 by 3+\sqrt{5}.

$$\left(12+4\sqrt{5}\right)\left(3-\sqrt{5}\right)-\left(\sqrt{5}-1\right)^{2}$$

Use the distributive property to multiply 12+4\sqrt{5} by 3-\sqrt{5} and combine like terms.

$$36-4\left(\sqrt{5}\right)^{2}-\left(\sqrt{5}-1\right)^{2}$$

The square of \sqrt{5} is 5.

$$36-4\times 5-\left(\sqrt{5}-1\right)^{2}$$

Multiply -4 and 5 to get -20.

$$36-20-\left(\sqrt{5}-1\right)^{2}$$

Subtract 20 from 36 to get 16.

$$16-\left(\sqrt{5}-1\right)^{2}$$

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

$$16-\left(\left(\sqrt{5}\right)^{2}-2\sqrt{5}+1\right)$$

The square of \sqrt{5} is 5.

$$16-\left(5-2\sqrt{5}+1\right)$$

Add 5 and 1 to get 6.

$$16-\left(6-2\sqrt{5}\right)$$

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

$$16-6+2\sqrt{5}$$

Subtract 6 from 16 to get 10.

$$10+2\sqrt{5}$$