Solve frac{3-sqrt{2}}{(1-sqrt{5}}

$$\frac{ 3- \sqrt{ 2 } }{ (1- \sqrt{ 5 } }$$

Evaluate

$$\frac{\sqrt{2}+\sqrt{10}-3\sqrt{5}-3}{4}\approx -1.282928177$$

Solution Steps

Rationalize the denominator of \frac{3-\sqrt{2}}{1-\sqrt{5}} by multiplying numerator and denominator by 1+\sqrt{5}.

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

Consider \left(1-\sqrt{5}\right)\left(1+\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}.

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

Square 1. Square \sqrt{5}.

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

Subtract 5 from 1 to get -4.

$$\frac{\left(3-\sqrt{2}\right)\left(1+\sqrt{5}\right)}{-4}$$

Apply the distributive property by multiplying each term of 3-\sqrt{2} by each term of 1+\sqrt{5}.

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

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{3+3\sqrt{5}-\sqrt{2}-\sqrt{10}}{-4}$$

Multiply both numerator and denominator by -1.

$$\frac{-3-3\sqrt{5}+\sqrt{2}+\sqrt{10}}{4}$$