Solve frac{sqrt{5}-2}{1-(sqrt{5}-2)}=

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

Evaluate

$$\frac{\sqrt{5}-1}{4}\approx 0.309016994$$

Solution Steps

To find the opposite of \sqrt{5}-2, find the opposite of each term.

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

The opposite of -2 is 2.

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

Add 1 and 2 to get 3.

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

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

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

Consider \left(3-\sqrt{5}\right)\left(3+\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}.

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

Square 3. Square \sqrt{5}.

$$\frac{\left(\sqrt{5}-2\right)\left(3+\sqrt{5}\right)}{9-5}$$

Subtract 5 from 9 to get 4.

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

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

$$\frac{3\sqrt{5}+\left(\sqrt{5}\right)^{2}-6-2\sqrt{5}}{4}$$

The square of \sqrt{5} is 5.

$$\frac{3\sqrt{5}+5-6-2\sqrt{5}}{4}$$

Subtract 6 from 5 to get -1.

$$\frac{3\sqrt{5}-1-2\sqrt{5}}{4}$$

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

$$\frac{\sqrt{5}-1}{4}$$