Solve frac{2-sqrt{5}}{5+sqrt{5}}

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

Evaluate

$$-\frac{7\sqrt{5}}{20}+\frac{3}{4}\approx -0.032623792$$

Solution Steps

Rationalize the denominator of \frac{2-\sqrt{5}}{5+\sqrt{5}} by multiplying numerator and denominator by 5-\sqrt{5}.

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

Consider \left(5+\sqrt{5}\right)\left(5-\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(2-\sqrt{5}\right)\left(5-\sqrt{5}\right)}{5^{2}-\left(\sqrt{5}\right)^{2}}$$

Square 5. Square \sqrt{5}.

$$\frac{\left(2-\sqrt{5}\right)\left(5-\sqrt{5}\right)}{25-5}$$

Subtract 5 from 25 to get 20.

$$\frac{\left(2-\sqrt{5}\right)\left(5-\sqrt{5}\right)}{20}$$

Apply the distributive property by multiplying each term of 2-\sqrt{5} by each term of 5-\sqrt{5}.

$$\frac{10-2\sqrt{5}-5\sqrt{5}+\left(\sqrt{5}\right)^{2}}{20}$$

Combine -2\sqrt{5} and -5\sqrt{5} to get -7\sqrt{5}.

$$\frac{10-7\sqrt{5}+\left(\sqrt{5}\right)^{2}}{20}$$

The square of \sqrt{5} is 5.

$$\frac{10-7\sqrt{5}+5}{20}$$

Add 10 and 5 to get 15.

$$\frac{15-7\sqrt{5}}{20}$$