Answer
$$(5+2i)^2+(2-i)^2=16i+24$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(5+2i)^2+(2-i)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}25+20i+4i^2+4-4i+i^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}25+20i-4+4-4i-1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}20i+21-4i+3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}16i+24\end{aligned} $$ | |
| ① | Find $ \left(5+2i\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 5 } $ and $ B = \color{red}{ 2i }$. $$ \begin{aligned}\left(5+2i\right)^2 = \color{blue}{5^2} +2 \cdot 5 \cdot 2i + \color{red}{\left( 2i \right)^2} = 25+20i+4i^2\end{aligned} $$Find $ \left(2-i\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 2 } $ and $ B = \color{red}{ i }$. $$ \begin{aligned}\left(2-i\right)^2 = \color{blue}{2^2} -2 \cdot 2 \cdot i + \color{red}{i^2} = 4-4i+i^2\end{aligned} $$ |
| ② | $$ 4i^2 = 4 \cdot (-1) = -4 $$$$ i^2 = -1 $$ |
| ③ | Combine like terms: $$ \color{blue}{25} +20i \color{blue}{-4} = 20i+ \color{blue}{21} $$Combine like terms: $$ \color{blue}{4} -4i \color{blue}{-1} = -4i+ \color{blue}{3} $$ |
| ④ | Combine like terms: $$ \color{blue}{20i} + \color{red}{21} \color{blue}{-4i} + \color{red}{3} = \color{blue}{16i} + \color{red}{24} $$ |
This page was created using
Complex Expression calculator