Question
Answer
$$(3-2i)\cdot(5+4i)-(3-4i)^2=26i+30$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(3-2i)\cdot(5+4i)-(3-4i)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(3-2i)\cdot(5+4i)-(9-24i+16i^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(3-2i)\cdot(5+4i)-(9-24i-16) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(3-2i)\cdot(5+4i)-(-24i-7) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}15+12i-10i-8i^2-(-24i-7) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-8i^2+2i+15-(-24i-7) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}8+2i+15-(-24i-7) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}2i+23-(-24i-7) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}2i+23+24i+7 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}26i+30\end{aligned} $$ | |
| ① | Find $ \left(3-4i\right)^2 $ using 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 }$. $$ \begin{aligned}\left(3-4i\right)^2 = \color{blue}{3^2} -2 \cdot 3 \cdot 4i + \color{red}{\left( 4i \right)^2} = 9-24i+16i^2\end{aligned} $$ |
| ② | $$ 16i^2 = 16 \cdot (-1) = -16 $$ |
| ③ | Combine like terms: $$ \color{blue}{9} -24i \color{blue}{-16} = -24i \color{blue}{-7} $$ |
| ④ | Multiply each term of $ \left( \color{blue}{3-2i}\right) $ by each term in $ \left( 5+4i\right) $. $$ \left( \color{blue}{3-2i}\right) \cdot \left( 5+4i\right) = 15+12i-10i-8i^2 $$ |
| ⑤ | Combine like terms: $$ 15+ \color{blue}{12i} \color{blue}{-10i} -8i^2 = -8i^2+ \color{blue}{2i} +15 $$ |
| ⑥ | $$ -8i^2 = -8 \cdot (-1) = 8 $$ |
| ⑦ | Combine like terms: $$ \color{blue}{8} +2i+ \color{blue}{15} = 2i+ \color{blue}{23} $$ |
| ⑧ | Remove the parentheses by changing the sign of each term within them. $$ - \left( -24i-7 \right) = 24i+7 $$ |
| ⑨ | Combine like terms: $$ \color{blue}{2i} + \color{red}{23} + \color{blue}{24i} + \color{red}{7} = \color{blue}{26i} + \color{red}{30} $$ |
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