Question
Answer
$$(2-3i)\cdot(1+5i)-3\cdot(-2+i)=4i+23$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(2-3i)\cdot(1+5i)-3\cdot(-2+i)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}2+10i-3i-15i^2-(-6+3i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}-15i^2+7i+2-(-6+3i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}15+7i+2-(-6+3i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}7i+17-(-6+3i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}7i+17+6-3i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}4i+23\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{2-3i}\right) $ by each term in $ \left( 1+5i\right) $. $$ \left( \color{blue}{2-3i}\right) \cdot \left( 1+5i\right) = 2+10i-3i-15i^2 $$Multiply $ \color{blue}{3} $ by $ \left( -2+i\right) $ $$ \color{blue}{3} \cdot \left( -2+i\right) = -6+3i $$ |
| ② | Combine like terms: $$ 2+ \color{blue}{10i} \color{blue}{-3i} -15i^2 = -15i^2+ \color{blue}{7i} +2 $$ |
| ③ | $$ -15i^2 = -15 \cdot (-1) = 15 $$ |
| ④ | Combine like terms: $$ \color{blue}{15} +7i+ \color{blue}{2} = 7i+ \color{blue}{17} $$ |
| ⑤ | Remove the parentheses by changing the sign of each term within them. $$ - \left( -6+3i \right) = 6-3i $$ |
| ⑥ | Combine like terms: $$ \color{blue}{7i} + \color{red}{17} + \color{red}{6} \color{blue}{-3i} = \color{blue}{4i} + \color{red}{23} $$ |
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