If `b_(1)b_(2) = 2(c_(1) + c_(2))`, then at least one of the equations `x^(2) + b_(1)x + c_(1) = 0` and `x^(2) + b_(2)x + c_(2) = 0` has

Options

A. imaginary roots
B. real roots
C. purely imaginary roots
D. none of these

Answer

2

Explanation

Let `D_(1) and D_(2)` be discriminants of `x^(2) + b_(1)x + c_(1) = 0`
and `x^(2) + b_(2)x + c_(2) = 0`, respectively. Then,
`D_(1) + D_(2) = b_(1)^(2) - 4c_(1) + b_(2)^(2) - 4c_(2)`
`= (b_(1)^(2) + b_(2)^(2)) - 4(c_(1) + c_(2))`
`= b_(1)^(2) + b_(2)^(2) - 2b_(1)b_(2)" "[because b_(1)b_(2) = 2(c_(1) + c_(2))]`
`=(b_(1) - b_(2))^(2) ge 0`
`rArr D_(1) ge 0` or `D_(2) ge 0` or `D_(1) and D_(2)` both are positive Hence, at least one of the equations has real roots.