Complex Numbers And Quadratic Equations question 813
Question: If a, b and g are the roots of $ x^{3}+8=0 $ , then the equation whose roots are $ {{\alpha }^{2}},{{\beta }^{2}} $ and $ {{\gamma }^{2}} $ is
Options:
A) $ x^{3}-8=0 $
B) $ x^{3}-16=0 $
C) $ x^{3}+64=0 $
D) $ x^{3}-64=0 $
Show Answer
Answer:
Correct Answer: D
Solution:
Let $ y=x^{2} $ . Then $ x=\sqrt{y} $ \ $ x^{3}+8=0\Rightarrow {y^{3/2}}+8=0 $
Þ $ y^{3}=64\Rightarrow y^{3}-64=0 $ Thus the equation having roots $ {{\alpha }^{2}},{{\beta }^{2}} $ and $ {{\gamma }^{2}} $ is $ x^{3}- ({{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}})x^{2}+ ({{\alpha }^{2}}{{\beta }^{2}}+{{\alpha }^{2}}{{\gamma }^{2}}+{{\beta }^{2}}{{\gamma }^{2}})x - {{\alpha }^{2}}{{\beta }^{2}}{{\gamma }^{2}} =0 $ .
BETA
