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Question

If a, b, c are rational numbers (a>b>c>0) and the quadratic equation (a+b2c)x2+(b+c2a)x+(c+a2b)=0 has a root in the interval (1,0), then which of the following statements is (are) CORRECT?
  1. a+c<2b
  2. Both roots are rational
  3. ax2+2bx+c=0 has both roots negative
  4. cx2+2bx+a=0 has both roots negative


Solution

The correct options are
A a+c<2b
B Both roots are rational
C ax2+2bx+c=0 has both roots negative
D cx2+2bx+a=0 has both roots negative
(a+b2c)x2+(b+c2a)x+(c+a2b)=0, a+b2c0
The given equation has one root as 1.
Let α be the other root.
1α=c+a2ba+b2c
So, the other root is x=c+a2ba+b2c 
As c+a2ba+b2c<0 and a+b>2c
c+a2b<0
a+c<2b

For equation ax2+2bx+c=0,
f(0)=c>0,2ba<0 
So both roots are negative.

Similarly, for cx2+2bx+a=0, both roots are negative.

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