Question
Answer
$$(-1+4i)^2=-8i-15$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(-1+4i)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}1-8i+16i^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}1-8i-16 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-8i-15\end{aligned} $$ | |
| ① | Find $ \left(-1+4i\right)^2 $ in two steps. S1: Change all signs inside bracket. S2: Apply 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}{ 4i }$. $$ \begin{aligned}\left(-1+4i\right)^2& \xlongequal{ S1 } \left(1-4i\right)^2 \xlongequal{ S2 } \color{blue}{1^2} -2 \cdot 1 \cdot 4i + \color{red}{\left( 4i \right)^2} = \\[1 em] & = 1-8i+16i^2\end{aligned} $$ |
| ② | $$ 16i^2 = 16 \cdot (-1) = -16 $$ |
| ③ | Combine like terms: $$ -8i+ \color{blue}{1} \color{blue}{-16} = -8i \color{blue}{-15} $$ |
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