Complex Numbers And Quadratic Equations question 536
Question: If one root of $ ax^{2}+bx+c=0 $ be square of the other, then the value of $ b^{3}+ac^{2}+a^{2}c $ is
Options:
A) $ 3abc $
B) $ -3abc $
C) 0
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
Let $ \alpha ,{{\alpha }^{2}} $ be the two roots. Then $ \alpha +{{\alpha }^{2}}=-\frac{b}{a} $ …..(i) and $ \alpha .{{\alpha }^{2}}=\frac{c}{a} $ …..(ii) On cubing both sides of (i) $ {{\alpha }^{3}}+{{\alpha }^{6}}+3\alpha {{\alpha }^{2}}(\alpha +{{\alpha }^{2}})=-\frac{b^{3}}{a^{3}} $
$ \Rightarrow \frac{c}{a}+\frac{c^{2}}{a^{2}}+3\frac{c}{a}( -\frac{b}{a} )=-\frac{b^{3}}{c^{3}} $ [By (i) and (ii)]
$ \Rightarrow b^{3}+ac^{2}+a^{2}c=3abc $ .
BETA
