Complex Numbers And Quadratic Equations question 580
Question: If one root of $ x^{2}-x-k=0 $ is square of the other, then k = [EAMCET 1986, 1987]
Options:
A) $ 2\pm \sqrt{3} $
B) $ 3\pm \sqrt{2} $
C) $ 2\pm \sqrt{5} $
D) $ 5\pm \sqrt{2} $
Show Answer
Answer:
Correct Answer: C
Solution:
Let $ \alpha $ and $ {{\alpha }^{2}} $ be the roots of $ x^{2}-x-k=0 $ . Then $ \alpha +{{\alpha }^{2}}=1 $ and $ {{\alpha }^{3}}=-k $ .
$ \therefore {{(-k)}^{1/3}}+{{(-k)}^{2/3}}=1\Rightarrow -{k^{1/3}}+{k^{2/3}}=1 $
Þ $ {{({k^{2/3}}-{k^{1/3}})}^{3}}=1\Rightarrow k^{2}-k-3k({k^{2/3}}-{k^{1/3}})=1 $
Þ $ k^{2}-k-3k(1)=1 $
Þ $ k^{2}-4k-1=0\Rightarrow k=2\pm \sqrt{5} $ .
BETA
