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

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

Evaluate

$$4\sqrt{5}-9\approx -0.05572809$$

Solution Steps

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

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

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

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

Square \sqrt{5}. Square 2.

$$\frac{\left(2-\sqrt{5}\right)\left(\sqrt{5}-2\right)}{5-4}$$

Subtract 4 from 5 to get 1.

$$\frac{\left(2-\sqrt{5}\right)\left(\sqrt{5}-2\right)}{1}$$

Anything divided by one gives itself.

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

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

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

The square of \sqrt{5} is 5.

$$2\sqrt{5}-4-5+2\sqrt{5}$$

Subtract 5 from -4 to get -9.

$$2\sqrt{5}-9+2\sqrt{5}$$

Combine 2\sqrt{5} and 2\sqrt{5} to get 4\sqrt{5}.

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