Solve frac{1+sqrt{5}}{sqrt{5}-2}

$$\frac { 1 + \sqrt { 5 } } { \sqrt { 5 } - 2 }$$

Evaluate

$$3\sqrt{5}+7\approx 13.708203932$$

Solution Steps

Rationalize the denominator of \frac{1+\sqrt{5}}{\sqrt{5}-2} by multiplying numerator and denominator by \sqrt{5}+2.

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

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}.

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

Square \sqrt{5}. Square 2.

$$\frac{\left(1+\sqrt{5}\right)\left(\sqrt{5}+2\right)}{5-4}$$

Subtract 4 from 5 to get 1.

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

Anything divided by one gives itself.

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

Apply the distributive property by multiplying each term of 1+\sqrt{5} by each term of \sqrt{5}+2.

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

The square of \sqrt{5} is 5.

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

Add 2 and 5 to get 7.

$$\sqrt{5}+7+2\sqrt{5}$$

Combine \sqrt{5} and 2\sqrt{5} to get 3\sqrt{5}.

$$3\sqrt{5}+7$$