Question
Answer
$$\dfrac{3}{-6+\sqrt{5}}=\dfrac{-18-3\sqrt{5}}{31}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3}{-6+\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3}{-6+\sqrt{5}}\frac{-6-\sqrt{5}}{-6-\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-18-3\sqrt{5}}{36+6\sqrt{5}-6\sqrt{5}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-18-3\sqrt{5}}{31}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ -6- \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 3 } \cdot \left( -6- \sqrt{5}\right) = \color{blue}{3} \cdot-6+\color{blue}{3} \cdot- \sqrt{5} = \\ = -18- 3 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( -6 + \sqrt{5}\right) } \cdot \left( -6- \sqrt{5}\right) = \color{blue}{-6} \cdot-6\color{blue}{-6} \cdot- \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot-6+\color{blue}{ \sqrt{5}} \cdot- \sqrt{5} = \\ = 36 + 6 \sqrt{5}- 6 \sqrt{5}-5 $$ |
| ③ | Simplify numerator and denominator |
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