Question
Answer
$$\dfrac{20(26i+1)^2}{26i((26i)^2+2\cdot26i+100)}=\dfrac{1287-97876i}{108706}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{20(26i+1)^2}{26i((26i)^2+2\cdot26i+100)}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{20(676i^2+52i+1)}{26i(676i^2+52i+100)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{20(-676+52i+1)}{26i(-676+52i+100)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{20(52i-675)}{26i(52i-576)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}\frac{1040i-13500}{1352i^2-14976i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}\frac{1040i-13500}{-1352-14976i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} } }}}\frac{1287-97876i}{108706}\end{aligned} $$ | |
| ① | Find $ \left(26i+1\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 26i } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(26i+1\right)^2 = \color{blue}{\left( 26i \right)^2} +2 \cdot 26i \cdot 1 + \color{red}{1^2} = 676i^2+52i+1\end{aligned} $$ |
| ② | $$ \left( 26i \right)^2 = 26^2i^2 = 676i^2 $$ |
| ③ | $$ 2 \cdot 26 i = 52 i $$ |
| ④ | $$ 676i^2 = 676 \cdot (-1) = -676 $$ |
| ⑤ | $$ 676i^2 = 676 \cdot (-1) = -676 $$ |
| ⑥ | Combine like terms: $$ \color{blue}{-676} +52i+ \color{blue}{1} = 52i \color{blue}{-675} $$ |
| ⑦ | Combine like terms: $$ \color{blue}{-676} +52i+ \color{blue}{100} = 52i \color{blue}{-576} $$ |
| ⑧ | Multiply $ \color{blue}{20} $ by $ \left( 52i-675\right) $ $$ \color{blue}{20} \cdot \left( 52i-675\right) = 1040i-13500 $$ |
| ⑨ | Multiply $ \color{blue}{26i} $ by $ \left( 52i-576\right) $ $$ \color{blue}{26i} \cdot \left( 52i-576\right) = 1352i^2-14976i $$ |
| ⑩ | $$ 1352i^2 = 1352 \cdot (-1) = -1352 $$ |
| ⑪ | Divide $ \, -13500+1040i \, $ by $ \, -1352-14976i \, $ to get $\,\, \dfrac{1287-97876i}{108706} $. ( view steps ) |
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