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

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

Evaluate

$$1-8\sqrt{5}\approx -16.88854382$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

Add 5 and 4 to get 9.

$$9-4\sqrt{5}-\left(\sqrt{5}+1\right)\left(\sqrt{5}+3\right)$$

Use the distributive property to multiply \sqrt{5}+1 by \sqrt{5}+3 and combine like terms.

$$9-4\sqrt{5}-\left(\left(\sqrt{5}\right)^{2}+4\sqrt{5}+3\right)$$

The square of \sqrt{5} is 5.

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

Add 5 and 3 to get 8.

$$9-4\sqrt{5}-\left(8+4\sqrt{5}\right)$$

To find the opposite of 8+4\sqrt{5}, find the opposite of each term.

$$9-4\sqrt{5}-8-4\sqrt{5}$$

Subtract 8 from 9 to get 1.

$$1-4\sqrt{5}-4\sqrt{5}$$

Combine -4\sqrt{5} and -4\sqrt{5} to get -8\sqrt{5}.

$$1-8\sqrt{5}$$