Answer
$$\frac{\sqrt{5}}{\sqrt{5}-5}=-\frac{1+\sqrt{5}}{4}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}}{\sqrt{5}-5}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}}{\sqrt{5}-5}\frac{\sqrt{5}+5}{\sqrt{5}+5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5+5\sqrt{5}}{5+5\sqrt{5}-5\sqrt{5}-25} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{5+5\sqrt{5}}{-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{1+\sqrt{5}}{-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{1+\sqrt{5}}{4}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{5} + 5} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{5} } \cdot \left( \sqrt{5} + 5\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot5 = \\ = 5 + 5 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{5}-5\right) } \cdot \left( \sqrt{5} + 5\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot5\color{blue}{-5} \cdot \sqrt{5}\color{blue}{-5} \cdot5 = \\ = 5 + 5 \sqrt{5}- 5 \sqrt{5}-25 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 5. |
| ⑤ | Place a negative sign in front of a fraction. |
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