lf the quadratic equations `x^2+bx+c=0` and `bx^2+cx+1=0` have a common root then prove that either `b+c+1=0` or `b^2+c^2+1=bc+b+c`.

Options

A. `b + c + 1=0`
B. `b^(2) + c^(2) -1 = bc - b - c`
C. `b + c - 1=0`
D. `b^(2) + c^(2) + 1 = bc + b +c`

Answer

1,4

Explanation

Equations `x^(2) + bx + c = 0` and `bx^(2) + cx + 1 = 0` have a common roots.
`(bc - 1)^(2) = (c-b^(2)) (b-c^(2))`
`rArr b^(2)c^(2) + 1- 2bc = bc - b^(3) - c^(3) + b^(2)c^(2)`
`rArr 1 + c^(3) + b^(3) = 3bc`
`rArr (1+ c+b) (c^(2) + b^(2) + 1-b-c- bc) = 0`
`rArr b + c + 1 = 0 and c^(2) + b^(2) + 1 = c + b + bc`