Complex Numbers And Quadratic Equations question 595
Question: The equation formed by decreasing each root of $ ax^{2}+bx+c=0 $ by 1 is $ 2x^{2}+8x+2=0, $ then [EAMCET 2000]
Options:
A) a = - b
B) b = - c
C) c = - a
D) b = a + c
Show Answer
Answer:
Correct Answer: B
Solution:
$ \alpha ,\beta $ be the roots of $ ax^{2}+bx+c=0 $
$ \therefore \alpha +\beta =-b/a,\alpha \beta =c/a $ Now roots are $ \alpha -1,\beta -1 $ Their sum, $ \alpha +\beta -2=(-b/a)-2=-8/2=-4 $ Their product, $ (\alpha -1)(\beta -1)=\alpha \beta -(\alpha +\beta )+1 $ $ =c/a+b/a+1=1 $ $ \because $ New equation is $ 2x^{2}+8x+2=0 $
$ \therefore b/a=2 $ i.e. $ b=2a $ , also $ c+b=0\Rightarrow b=-c $ .
BETA
