Question
Answer
$$(-5+10i)^2=-100i-75$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(-5+10i)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}25-100i+100i^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}25-100i-100 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-100i-75\end{aligned} $$ | |
| ① | Find $ \left(-5+10i\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}{ 5 } $ and $ B = \color{red}{ 10i }$. $$ \begin{aligned}\left(-5+10i\right)^2& \xlongequal{ S1 } \left(5-10i\right)^2 \xlongequal{ S2 } \color{blue}{5^2} -2 \cdot 5 \cdot 10i + \color{red}{\left( 10i \right)^2} = \\[1 em] & = 25-100i+100i^2\end{aligned} $$ |
| ② | $$ 100i^2 = 100 \cdot (-1) = -100 $$ |
| ③ | Combine like terms: $$ -100i+ \color{blue}{25} \color{blue}{-100} = -100i \color{blue}{-75} $$ |
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