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

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

Evaluate

$$5$$

Solution Steps

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

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

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}.

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

Square \sqrt{5}. Square 2.

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

Subtract 4 from 5 to get 1.

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

Anything divided by one gives itself.

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

Use the distributive property to multiply \sqrt{5} by \sqrt{5}+2.

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

The square of \sqrt{5} is 5.

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

Combine 2\sqrt{5} and -2\sqrt{5} to get 0.

$$5$$

Factor

$$5$$