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

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

Evaluate

$$10\sqrt{5}-5\approx 17.360679775$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

Add 5 and 4 to get 9.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-\sqrt{5}\right)^{2}.

$$9+4\sqrt{5}-\left(9-6\sqrt{5}+\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$9+4\sqrt{5}-\left(9-6\sqrt{5}+5\right)$$

Add 9 and 5 to get 14.

$$9+4\sqrt{5}-\left(14-6\sqrt{5}\right)$$

To find the opposite of 14-6\sqrt{5}, find the opposite of each term.

$$9+4\sqrt{5}-14+6\sqrt{5}$$

Subtract 14 from 9 to get -5.

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

Combine 4\sqrt{5} and 6\sqrt{5} to get 10\sqrt{5}.

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

Expand

$$10 \sqrt{5} - 5 = 17.360679775$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

Add 5 and 4 to get 9.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-\sqrt{5}\right)^{2}.

$$9+4\sqrt{5}-\left(9-6\sqrt{5}+\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$9+4\sqrt{5}-\left(9-6\sqrt{5}+5\right)$$

Add 9 and 5 to get 14.

$$9+4\sqrt{5}-\left(14-6\sqrt{5}\right)$$

To find the opposite of 14-6\sqrt{5}, find the opposite of each term.

$$9+4\sqrt{5}-14+6\sqrt{5}$$

Subtract 14 from 9 to get -5.

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

Combine 4\sqrt{5} and 6\sqrt{5} to get 10\sqrt{5}.

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