$$\dfrac{-1-2i-(1+i)\dfrac{254-113i}{205}}{-3-2i}=\dfrac{2818+509i}{2665}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{-1-2i-(1+i)\frac{254-113i}{205}}{-3-2i}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{-1-2i-\frac{-113i^2+141i+254}{205}}{-3-2i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-1-2i-\frac{113+141i+254}{205}}{-3-2i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-1-2i-\frac{141i+367}{205}}{-3-2i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\frac{-551i-572}{205}}{-3-2i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-551i-572}{-410i-615} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{2818+509i}{2665}\end{aligned} $$ | |
| ① | Multiply $1+i$ by $ \dfrac{254-113i}{205} $ to get $ \dfrac{-113i^2+141i+254}{205} $. Step 1: Write $ 1+i $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} 1+i \cdot \frac{254-113i}{205} & \xlongequal{\text{Step 1}} \frac{1+i}{\color{red}{1}} \cdot \frac{254-113i}{205} \xlongequal{\text{Step 2}} \frac{ \left( 1+i \right) \cdot \left( 254-113i \right) }{ 1 \cdot 205 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 254-113i+254i-113i^2 }{ 205 } = \frac{-113i^2+141i+254}{205} \end{aligned} $$ |
| ② | $$ -113i^2 = -113 \cdot (-1) = 113 $$ |
| ③ | Combine like terms: $$ \color{blue}{113} +141i+ \color{blue}{254} = 141i+ \color{blue}{367} $$ |
| ④ | Subtract $ \dfrac{141i+367}{205} $ from $ -1-2i $ to get $ \dfrac{ \color{purple}{ -551i-572 } }{ 205 }$. Step 1: Write $ -1-2i $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: To subtract raitonal expressions, both fractions must have the same denominator. |
| ⑤ | Divide $ \dfrac{-551i-572}{205} $ by $ -3-2i $ to get $ \dfrac{-551i-572}{-410i-615} $. Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{ \frac{-551i-572}{205} }{-3-2i} & \xlongequal{\text{Step 1}} \frac{-551i-572}{205} \cdot \frac{\color{blue}{1}}{\color{blue}{-3-2i}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( -551i-572 \right) \cdot 1 }{ 205 \cdot \left( -3-2i \right) } \xlongequal{\text{Step 3}} \frac{ -551i-572 }{ -615-410i } = \\[1ex] &= \frac{-551i-572}{-410i-615} \end{aligned} $$ |
| ⑥ | Divide $ \, -572-551i \, $ by $ \, -615-410i \, $ to get $\,\, \dfrac{2818+509i}{2665} $. ( view steps ) |