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Question

Answer

$\frac{10 ( - 3 + 4 i ) ^{2} + 50}{2 ( - 3 + 4 i ) ^{2} + 5 \cdot ( - 3 + 4 i ) + 10} = \frac{1420 + 800 i}{229}$

Explanation

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$\frac{10 ( - 3 + 4 i ) ^{2} + 50}{2 ( - 3 + 4 i ) ^{2} + 5 \cdot ( - 3 + 4 i ) + 10} \overset{◯}{1} \overset{◯}{2} \frac{10 ( 9 - 24 i + 16 i ^{2} ) + 50}{2 ( 9 - 24 i + 16 i ^{2} ) + 5 \cdot ( - 3 + 4 i ) + 10} \overset{◯}{3} \overset{◯}{4} \frac{10 ( 9 - 24 i - 16 ) + 50}{2 ( 9 - 24 i - 16 ) + 5 \cdot ( - 3 + 4 i ) + 10} \overset{◯}{5} \overset{◯}{6} \frac{10 ( - 24 i - 7 ) + 50}{2 ( - 24 i - 7 ) + 5 \cdot ( - 3 + 4 i ) + 10} \overset{◯}{7} \overset{◯}{8} \overset{◯}{9} \frac{- 240 i - 70 + 50}{- 48 i - 14 - 15 + 20 i + 10} \overset{◯}{10} \overset{◯}{11} \frac{- 240 i - 70 + 50}{- 28 i - 29 + 10} \overset{◯}{12} \frac{- 240 i - 20}{- 28 i - 29 + 10} \overset{◯}{13} \frac{- 240 i - 20}{- 28 i - 19} \overset{◯}{14} \frac{1420 + 800 i}{229}$
①	Find $(- 3 + 4 i)^{2}$ in two steps. S1: Change all signs inside bracket. S2: Apply formula $(A - B)^{2} = A^{2} - 2 \cdot A \cdot B + B^{2}$ where $A = 3$ and $B = 4 i$ . $(- 3 + 4 i)^{2} S 1 (3 - 4 i)^{2} S 2 3^{2} - 2 \cdot 3 \cdot 4 i + (4 i)^{2} = = 9 - 24 i + 16 i^{2}$
②	Find $(- 3 + 4 i)^{2}$ in two steps. S1: Change all signs inside bracket. S2: Apply formula $(A - B)^{2} = A^{2} - 2 \cdot A \cdot B + B^{2}$ where $A = 3$ and $B = 4 i$ . $(- 3 + 4 i)^{2} S 1 (3 - 4 i)^{2} S 2 3^{2} - 2 \cdot 3 \cdot 4 i + (4 i)^{2} = = 9 - 24 i + 16 i^{2}$
③	$16 i^{2} = 16 \cdot (- 1) = - 16$
④	$16 i^{2} = 16 \cdot (- 1) = - 16$
⑤	Combine like terms: $9 - 24 i - 16 = - 24 i - 7$
⑥	Combine like terms: $9 - 24 i - 16 = - 24 i - 7$
⑦	Multiply $10$ by $(- 24 i - 7)$ $10 \cdot (- 24 i - 7) = - 240 i - 70$
⑧	Multiply $2$ by $(- 24 i - 7)$ $2 \cdot (- 24 i - 7) = - 48 i - 14$
⑨	Multiply $5$ by $(- 3 + 4 i)$ $5 \cdot (- 3 + 4 i) = - 15 + 20 i$
⑩	Multiply $10$ by $(- 24 i - 7)$ $10 \cdot (- 24 i - 7) = - 240 i - 70$
⑪	Combine like terms: $- 48 i - 14 - 15 + 20 i = - 28 i - 29$
⑫	Simplify numerator $- 240 i - 70 + 50 = - 240 i - 20$
⑬	Simplify denominator $- 28 i - 29 + 10 = - 28 i - 19$
⑭	Divide $- 20 - 240 i$ by $- 19 - 28 i$ to get $\frac{1420 + 800 i}{229}$ . ( view steps )

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