Complex Numbers And Quadratic Equations question 673
Question: If $ 2x=3+5i, $ then what is the value of $ 2x^{3}+2x^{2}-7x+72? $
Options:
A) 4
B) $ -4 $
C) 8
D) $ -8 $
Show Answer
Answer:
Correct Answer: A
Solution:
Given $ 2x=3+5i $
$ \Rightarrow x=\frac{3+5i}{2} $
Consider $x^{3} =\frac{27+125i^{3}+225i^{2}+135i}{8} $ $ =\frac{27 - 125i - 225 +135i}{8}( \begin{matrix} \because i^{2}=1 \\ i^{3}=-i \\ \end{matrix} ) $ $ =\frac{-198+10i}{8}=\frac{-99+5i}{4} $ and $ x^{2}=\frac{9+25i^{2}+30i}{4} $ $ =\frac{9-25+30i}{4}=\frac{-8+15i}{2} $
Now, Consider $ 2x^{3} + 2x^{2}- 7x + 72 $
$ =( \frac{-99+5i}{2} )+(-8+15i)-\frac{7(3+5i)}{2}9+72 $
$ =-\frac{99}{2}+\frac{5i}{2}-8+15i-\frac{21}{2}-\frac{35}{2}i+72 $
$ =( -\frac{99}{2}-8-\frac{21}{2}+72 )+( \frac{5}{2}+15-\frac{35}{2} )i $ $ =\frac{-99-16-21+144}{2}=\frac{8}{2}=4 $
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