Question
Answer
$$\dfrac{10\cdot(-3+4i)+50}{2(-3+4i)^2+5\cdot(-3+4i)+10}=\dfrac{-300-40i}{229}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{10\cdot(-3+4i)+50}{2(-3+4i)^2+5\cdot(-3+4i)+10}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-30+40i+50}{2(9-24i+16i^2)+5\cdot(-3+4i)+10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-30+40i+50}{2(9-24i-16)+5\cdot(-3+4i)+10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{-30+40i+50}{2(-24i-7)+5\cdot(-3+4i)+10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}\frac{-30+40i+50}{-48i-14-15+20i+10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} } }}}\frac{-30+40i+50}{-28i-29+10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle12}{\textcircled {12}} } }}}\frac{40i+20}{-28i-29+10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle13}{\textcircled {13}} } }}}\frac{40i+20}{-28i-19} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle14}{\textcircled {14}} } }}}\frac{-300-40i}{229}\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{10} $ by $ \left( -3+4i\right) $ $$ \color{blue}{10} \cdot \left( -3+4i\right) = -30+40i $$ |
| ② | Find $ \left(-3+4i\right)^2 $ in two steps. S1: Change all signs inside bracket. S2: Apply formula $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 3 } $ and $ B = \color{red}{ 4i }$. $$ \begin{aligned}\left(-3+4i\right)^2& \xlongequal{ S1 } \left(3-4i\right)^2 \xlongequal{ S2 } \color{blue}{3^2} -2 \cdot 3 \cdot 4i + \color{red}{\left( 4i \right)^2} = \\[1 em] & = 9-24i+16i^2\end{aligned} $$ |
| ③ | Multiply $ \color{blue}{10} $ by $ \left( -3+4i\right) $ $$ \color{blue}{10} \cdot \left( -3+4i\right) = -30+40i $$ |
| ④ | $$ 16i^2 = 16 \cdot (-1) = -16 $$ |
| ⑤ | Multiply $ \color{blue}{10} $ by $ \left( -3+4i\right) $ $$ \color{blue}{10} \cdot \left( -3+4i\right) = -30+40i $$ |
| ⑥ | Combine like terms: $$ \color{blue}{9} -24i \color{blue}{-16} = -24i \color{blue}{-7} $$ |
| ⑦ | Multiply $ \color{blue}{10} $ by $ \left( -3+4i\right) $ $$ \color{blue}{10} \cdot \left( -3+4i\right) = -30+40i $$ |
| ⑧ | Multiply $ \color{blue}{2} $ by $ \left( -24i-7\right) $ $$ \color{blue}{2} \cdot \left( -24i-7\right) = -48i-14 $$ |
| ⑨ | Multiply $ \color{blue}{5} $ by $ \left( -3+4i\right) $ $$ \color{blue}{5} \cdot \left( -3+4i\right) = -15+20i $$ |
| ⑩ | Multiply $ \color{blue}{10} $ by $ \left( -3+4i\right) $ $$ \color{blue}{10} \cdot \left( -3+4i\right) = -30+40i $$ |
| ⑪ | Combine like terms: $$ \color{blue}{-48i} \color{red}{-14} \color{red}{-15} + \color{blue}{20i} = \color{blue}{-28i} \color{red}{-29} $$ |
| ⑫ | Simplify numerator $$ \color{blue}{-30} +40i+ \color{blue}{50} = 40i+ \color{blue}{20} $$ |
| ⑬ | Simplify denominator $$ -28i \color{blue}{-29} + \color{blue}{10} = -28i \color{blue}{-19} $$ |
| ⑭ | Divide $ \, 20+40i \, $ by $ \, -19-28i \, $ to get $\,\, \dfrac{-300-40i}{229} $. ( view steps ) |
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