Answer
$$\frac{2}{-1+\sqrt{5}}=\frac{1+\sqrt{5}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2}{-1+\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2}{-1+\sqrt{5}}\frac{-1-\sqrt{5}}{-1-\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-2-2\sqrt{5}}{1+\sqrt{5}-\sqrt{5}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-2-2\sqrt{5}}{-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-1-\sqrt{5}}{-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{1+\sqrt{5}}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ -1- \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 2 } \cdot \left( -1- \sqrt{5}\right) = \color{blue}{2} \cdot-1+\color{blue}{2} \cdot- \sqrt{5} = \\ = -2- 2 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( -1 + \sqrt{5}\right) } \cdot \left( -1- \sqrt{5}\right) = \color{blue}{-1} \cdot-1\color{blue}{-1} \cdot- \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot-1+\color{blue}{ \sqrt{5}} \cdot- \sqrt{5} = \\ = 1 + \sqrt{5}- \sqrt{5}-5 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
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