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

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

Evaluate

$$6-5\sqrt{5}\approx -5.180339887$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

$$5-4\sqrt{5}+4-\left(3+\sqrt{5}\right)$$

Add 5 and 4 to get 9.

$$9-4\sqrt{5}-\left(3+\sqrt{5}\right)$$

To find the opposite of 3+\sqrt{5}, find the opposite of each term.

$$9-4\sqrt{5}-3-\sqrt{5}$$

Subtract 3 from 9 to get 6.

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

Combine -4\sqrt{5} and -\sqrt{5} to get -5\sqrt{5}.

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