Question
Answer
$$\dfrac{1}{(1+2i)^2}=\dfrac{1}{4i-3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{(1+2i)^2}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{1+4i+4i^2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{1}{1+4i-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{1}{4i-3}\end{aligned} $$ | |
| ① | Find $ \left(1+2i\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 1 } $ and $ B = \color{red}{ 2i }$. $$ \begin{aligned}\left(1+2i\right)^2 = \color{blue}{1^2} +2 \cdot 1 \cdot 2i + \color{red}{\left( 2i \right)^2} = 1+4i+4i^2\end{aligned} $$ |
| ② | $$ 4i^2 = 4 \cdot (-1) = -4 $$ |
| ③ | Simplify denominator $$ \color{blue}{1} +4i \color{blue}{-4} = 4i \color{blue}{-3} $$ |
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