Complex Numbers And Quadratic Equations question 689
Question: If $ \omega $ is a complex cube root of unity and $ x={{\omega }^{2}}-\omega -2, $ then what is the value of $ x^{2}+4x+7 $ ?
Options:
A) $ -2 $
B) $ -1 $
C) $ 0 $
D) $ 1 $
Show Answer
Answer:
Correct Answer: C
Solution:
Given $ x={{\omega }^{2}}-\omega -2 $
$ \Rightarrow x+2={{\omega }^{2}}-\omega $ On squaring both sides, we get $ (x+2)^2 = (\omega^2 - \omega)^2 $
$ \Rightarrow {{(x+2)}^{2}}={{( {{\omega }^{2}}-\omega )}^{2}} $
$ \Rightarrow {x^{2}} + 4x + 4 = {{\omega }^{4}} + {{\omega }^{2}}- 2 {{\omega }^{3}} $ Add 3 to both sides
$ \Rightarrow {x^{2}}+4x+4+3=\omega +{{\omega }^{2}}-2+3( \because {{\omega }^{3}}=1 ) $ $ {x^{2}} + 4x + 7 = 1 + \omega + {{\omega }^{2}} $
$ \Rightarrow {x^{2}} +16x -7 =0( \because 1+\omega +{{\omega }^{2}} =0 )$
BETA
