Solve {left(sqrt{5}-1right)}^2+5(sqrt{5}-1)-6

$${ \left( \sqrt{ 5 } -1 \right) }^{ 2 } +5( \sqrt{ 5 } -1)-6$$

Evaluate

$$3\sqrt{5}-5\approx 1.708203932$$

Solution Steps

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

$$\left(\sqrt{5}\right)^{2}-2\sqrt{5}+1+5\left(\sqrt{5}-1\right)-6$$

The square of \sqrt{5} is 5.

$$5-2\sqrt{5}+1+5\left(\sqrt{5}-1\right)-6$$

Add 5 and 1 to get 6.

$$6-2\sqrt{5}+5\left(\sqrt{5}-1\right)-6$$

Use the distributive property to multiply 5 by \sqrt{5}-1.

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

Combine -2\sqrt{5} and 5\sqrt{5} to get 3\sqrt{5}.

$$6+3\sqrt{5}-5-6$$

Subtract 5 from 6 to get 1.

$$1+3\sqrt{5}-6$$

Subtract 6 from 1 to get -5.

$$-5+3\sqrt{5}$$