Solve frac{3-sqrt{5}}{sqrt{5}-1}

$$\frac{ 3- \sqrt{ 5 } }{ \sqrt{ 5 } -1 }$$

Evaluate

$$\frac{\sqrt{5}-1}{2}\approx 0.618033989$$

Solution Steps

Rationalize the denominator of \frac{3-\sqrt{5}}{\sqrt{5}-1} by multiplying numerator and denominator by \sqrt{5}+1.

$$\frac{\left(3-\sqrt{5}\right)\left(\sqrt{5}+1\right)}{\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)}$$

Consider \left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(3-\sqrt{5}\right)\left(\sqrt{5}+1\right)}{\left(\sqrt{5}\right)^{2}-1^{2}}$$

Square \sqrt{5}. Square 1.

$$\frac{\left(3-\sqrt{5}\right)\left(\sqrt{5}+1\right)}{5-1}$$

Subtract 1 from 5 to get 4.

$$\frac{\left(3-\sqrt{5}\right)\left(\sqrt{5}+1\right)}{4}$$

Apply the distributive property by multiplying each term of 3-\sqrt{5} by each term of \sqrt{5}+1.

$$\frac{3\sqrt{5}+3-\left(\sqrt{5}\right)^{2}-\sqrt{5}}{4}$$

The square of \sqrt{5} is 5.

$$\frac{3\sqrt{5}+3-5-\sqrt{5}}{4}$$

Subtract 5 from 3 to get -2.

$$\frac{3\sqrt{5}-2-\sqrt{5}}{4}$$

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

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