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

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

Evaluate

$$-20\sqrt{5}\approx -44.72135955$$

Solution Steps

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

$$25-10\sqrt{5}+\left(\sqrt{5}\right)^{2}-\left(5+\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$25-10\sqrt{5}+5-\left(5+\sqrt{5}\right)^{2}$$

Add 25 and 5 to get 30.

$$30-10\sqrt{5}-\left(5+\sqrt{5}\right)^{2}$$

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

$$30-10\sqrt{5}-\left(25+10\sqrt{5}+\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$30-10\sqrt{5}-\left(25+10\sqrt{5}+5\right)$$

Add 25 and 5 to get 30.

$$30-10\sqrt{5}-\left(30+10\sqrt{5}\right)$$

To find the opposite of 30+10\sqrt{5}, find the opposite of each term.

$$30-10\sqrt{5}-30-10\sqrt{5}$$

Subtract 30 from 30 to get 0.

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

Combine -10\sqrt{5} and -10\sqrt{5} to get -20\sqrt{5}.

$$-20\sqrt{5}$$

Expand

$$-20\sqrt{5}$$

Solution Steps

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

$$25-10\sqrt{5}+\left(\sqrt{5}\right)^{2}-\left(5+\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$25-10\sqrt{5}+5-\left(5+\sqrt{5}\right)^{2}$$

Add 25 and 5 to get 30.

$$30-10\sqrt{5}-\left(5+\sqrt{5}\right)^{2}$$

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

$$30-10\sqrt{5}-\left(25+10\sqrt{5}+\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$30-10\sqrt{5}-\left(25+10\sqrt{5}+5\right)$$

Add 25 and 5 to get 30.

$$30-10\sqrt{5}-\left(30+10\sqrt{5}\right)$$

To find the opposite of 30+10\sqrt{5}, find the opposite of each term.

$$30-10\sqrt{5}-30-10\sqrt{5}$$

Subtract 30 from 30 to get 0.

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

Combine -10\sqrt{5} and -10\sqrt{5} to get -20\sqrt{5}.

$$-20\sqrt{5}$$