Tap the blue circles to see an explanation.
$$ \begin{aligned}\frac{i(2i+1)+1}{2i+1+3i}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2i^2+i+1}{2i+1+3i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-2+i+1}{2i+1+3i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{i-1}{2i+1+3i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{i-1}{5i+1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{2+3i}{13}\end{aligned} $$ | |
① | Multiply $ \color{blue}{i} $ by $ \left( 2i+1\right) $ $$ \color{blue}{i} \cdot \left( 2i+1\right) = 2i^2+i $$ |
② | $$ 2i^2 = 2 \cdot (-1) = -2 $$ |
③ | $$ \color{blue}{-2} +i+ \color{blue}{1} = i \color{blue}{-1} $$ |
④ | $$ \color{blue}{2i} +1+ \color{blue}{3i} = \color{blue}{5i} +1 $$ |
⑤ | Divide $ \, -1+i \, $ by $ \, 1+5i \, $ to get $\,\, \dfrac{2+3i}{13} $. ( view steps ) |