Question
Answer
$$\dfrac{1-4i-(2+4i)\cdot(-3-i)}{4-i}=\dfrac{2+43i}{17}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1-4i-(2+4i)\cdot(-3-i)}{4-i}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1-4i-(-6-2i-12i-4i^2)}{4-i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{1-4i-(-4i^2-14i-6)}{4-i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{1-4i-(4-14i-6)}{4-i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{1-4i-(-14i-2)}{4-i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{1-4i+14i+2}{4-i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{10i+3}{4-i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{2+43i}{17}\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{2+4i}\right) $ by each term in $ \left( -3-i\right) $. $$ \left( \color{blue}{2+4i}\right) \cdot \left( -3-i\right) = -6-2i-12i-4i^2 $$ |
| ② | Combine like terms: $$ -6 \color{blue}{-2i} \color{blue}{-12i} -4i^2 = -4i^2 \color{blue}{-14i} -6 $$ |
| ③ | $$ -4i^2 = -4 \cdot (-1) = 4 $$ |
| ④ | Combine like terms: $$ \color{blue}{4} -14i \color{blue}{-6} = -14i \color{blue}{-2} $$ |
| ⑤ | Remove the parentheses by changing the sign of each term within them. $$ - \left( -14i-2 \right) = 14i+2 $$ |
| ⑥ | Simplify numerator $$ \color{blue}{1} \color{red}{-4i} + \color{red}{14i} + \color{blue}{2} = \color{red}{10i} + \color{blue}{3} $$ |
| ⑦ | Divide $ \, 3+10i \, $ by $ \, 4-i \, $ to get $\,\, \dfrac{2+43i}{17} $. ( view steps ) |
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