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Question

Answer

$$\dfrac{(2+3i)\cdot(1-2i)}{1-4i}+\dfrac{4+i}{2+3i}=\dfrac{233i+343}{221}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}\frac{(2+3i)\cdot(1-2i)}{1-4i}+\frac{4+i}{2+3i}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{(2+3i)\cdot(1-2i)}{1-4i}+\frac{11-10i}{13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2-4i+3i-6i^2}{1-4i}+\frac{11-10i}{13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-6i^2-i+2}{1-4i}+\frac{11-10i}{13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{6-i+2}{1-4i}+\frac{11-10i}{13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-i+8}{1-4i}+\frac{11-10i}{13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{12+31i}{17}+\frac{11-10i}{13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{233i+343}{221}\end{aligned} $$
①
Divide $ \, 4+i \, $ by $ \, 2+3i \, $ to get $\,\, \dfrac{11-10i}{13} $.
( view steps )
②

Multiply each term of $ \left( \color{blue}{2+3i}\right) $ by each term in $ \left( 1-2i\right) $.

$$ \left( \color{blue}{2+3i}\right) \cdot \left( 1-2i\right) = 2-4i+3i-6i^2 $$
③

Combine like terms:

$$ 2 \color{blue}{-4i} + \color{blue}{3i} -6i^2 = -6i^2 \color{blue}{-i} +2 $$
④
$$ -6i^2 = -6 \cdot (-1) = 6 $$
⑤

Combine like terms:

$$ \color{blue}{6} -i+ \color{blue}{2} = -i+ \color{blue}{8} $$
⑥
Divide $ \, 8-i \, $ by $ \, 1-4i \, $ to get $\,\, \dfrac{12+31i}{17} $.
( view steps )
⑦

Add $ \dfrac{12+31i}{17} $ and $ \dfrac{11-10i}{13} $ to get $ \dfrac{ \color{purple}{ 233i+343 } }{ 221 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 13 }$ and the second by $\color{blue}{ 17 }$.

$$ \begin{aligned} \frac{12+31i}{17} + \frac{11-10i}{13} & = \frac{ \left( 12+31i \right) \cdot \color{blue}{ 13 }}{ 17 \cdot \color{blue}{ 13 }} + \frac{ \left( 11-10i \right) \cdot \color{blue}{ 17 }}{ 13 \cdot \color{blue}{ 17 }} = \\[1ex] &=\frac{ \color{purple}{ 156+403i } }{ 221 } + \frac{ \color{purple}{ 187-170i } }{ 221 }=\frac{ \color{purple}{ 233i+343 } }{ 221 } \end{aligned} $$
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