If `alpha` and `beta`, `alpha` and `gamma`, `alpha` and `delta` are the roots of the equations `ax^(2)+2bx+c=0`, `2bx^(2)+cx+a=0` and `cx^(2)+ax+2b=0` respectively where `a`, `b`, `c` are positive real numbers, then `alpha+alpha^(2)` is equal to

Options

A. abc
B. a + 2b + c
C. -1
D. 0

Answer

3

Explanation

Since `alpha` is root of all equations
`aalpha^(2) + 2balpha + c = 0`
`2balpha^(2) + calpha + a = 0`
`calpha^(2) + aalpha + 2b = 0`
Adding we get `(a + 2b + c)(alpha^(2) + alpha + 1) = 0`
`a + 2b + c ne 0` as `a, b, c gt 0`
`rArr alpha^(2) + alpha + 1 = 0` or `alpha^(2) + alpha = -1`