Answer
$$\frac{2+\sqrt{5}}{4-2\sqrt{5}}=-\frac{9+4\sqrt{5}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2+\sqrt{5}}{4-2\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2+\sqrt{5}}{4-2\sqrt{5}}\frac{4+2\sqrt{5}}{4+2\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{8+4\sqrt{5}+4\sqrt{5}+10}{16+8\sqrt{5}-8\sqrt{5}-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{18+8\sqrt{5}}{-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{9+4\sqrt{5}}{-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{9+4\sqrt{5}}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 + 2 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 + \sqrt{5}\right) } \cdot \left( 4 + 2 \sqrt{5}\right) = \color{blue}{2} \cdot4+\color{blue}{2} \cdot 2 \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot4+\color{blue}{ \sqrt{5}} \cdot 2 \sqrt{5} = \\ = 8 + 4 \sqrt{5} + 4 \sqrt{5} + 10 $$ Simplify denominator. $$ \color{blue}{ \left( 4- 2 \sqrt{5}\right) } \cdot \left( 4 + 2 \sqrt{5}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot 2 \sqrt{5}\color{blue}{- 2 \sqrt{5}} \cdot4\color{blue}{- 2 \sqrt{5}} \cdot 2 \sqrt{5} = \\ = 16 + 8 \sqrt{5}- 8 \sqrt{5}-20 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Place a negative sign in front of a fraction. |
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