Question
Answer
$$\dfrac{-2+i-(-1-5i)\cdot(5-4i)}{4-3i}=\dfrac{26+157i}{25}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{-2+i-(-1-5i)\cdot(5-4i)}{4-3i}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{-2+i-(-5+4i-25i+20i^2)}{4-3i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-2+i-(20i^2-21i-5)}{4-3i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-2+i-(-20-21i-5)}{4-3i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-2+i-(-21i-25)}{4-3i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-2+i+21i+25}{4-3i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{22i+23}{4-3i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{26+157i}{25}\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{-1-5i}\right) $ by each term in $ \left( 5-4i\right) $. $$ \left( \color{blue}{-1-5i}\right) \cdot \left( 5-4i\right) = -5+4i-25i+20i^2 $$ |
| ② | Combine like terms: $$ -5+ \color{blue}{4i} \color{blue}{-25i} +20i^2 = 20i^2 \color{blue}{-21i} -5 $$ |
| ③ | $$ 20i^2 = 20 \cdot (-1) = -20 $$ |
| ④ | Combine like terms: $$ \color{blue}{-20} -21i \color{blue}{-5} = -21i \color{blue}{-25} $$ |
| ⑤ | Remove the parentheses by changing the sign of each term within them. $$ - \left( -21i-25 \right) = 21i+25 $$ |
| ⑥ | Simplify numerator $$ \color{blue}{-2} + \color{red}{i} + \color{red}{21i} + \color{blue}{25} = \color{red}{22i} + \color{blue}{23} $$ |
| ⑦ | Divide $ \, 23+22i \, $ by $ \, 4-3i \, $ to get $\,\, \dfrac{26+157i}{25} $. ( view steps ) |
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