Solve 2/3-sqrt{5}

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

Evaluate

$$\frac{\sqrt{5}+3}{2}\approx 2.618033989$$

Solution Steps

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

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

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}.

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

Square 3. Square \sqrt{5}.

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

Subtract 5 from 9 to get 4.

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

Divide 2\left(3+\sqrt{5}\right) by 4 to get \frac{1}{2}\left(3+\sqrt{5}\right).

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

Use the distributive property to multiply \frac{1}{2} by 3+\sqrt{5}.

$$\frac{1}{2}\times 3+\frac{1}{2}\sqrt{5}$$

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

$$\frac{3}{2}+\frac{1}{2}\sqrt{5}$$