Complex Numbers And Quadratic Equations question 588
Question: If $ \alpha $ and $ \beta $ are roots of the equation $ Ax^{2}+Bx+C=0 $ , then value of $ {{\alpha }^{3}}+{{\beta }^{3}} $ is [RPET 1996; DCE 2005]
Options:
A) $ \frac{3ABC-B^{3}}{A^{3}} $
B) $ \frac{3ABC+B^{3}}{A^{3}} $
C) $ \frac{B^{3}-3ABC}{A^{3}} $
D) $ \frac{B^{3}-3ABC}{B^{3}} $
Show Answer
Answer:
Correct Answer: A
Solution:
Given equation, $ Ax^{2}+Bx+C=0 $
Þ $ \alpha +\beta =-\frac{B}{A},\alpha \beta =\frac{C}{A} $ \ $ {{\alpha }^{3}}+{{\beta }^{3}}={{(\alpha +\beta )}^{3}}-3\alpha \beta (\alpha +\beta ) $ $ ={{( -\frac{B}{A} )}^{3}}-3( \frac{C}{A} )( -\frac{B}{A} ) $ $ =-\frac{B^{3}}{A^{3}}+\frac{3BC}{A^{2}} $ $ {{\alpha }^{3}}+{{\beta }^{3}}=\frac{3ABC-B^{3}}{A^{3}} $ .
BETA
