Let `f(x)=a x^2+bx +c `a ,b ,c` in R`. If `f(x)` takes real values for real values of `x` and non-real values for non-real values of `x` , then (a)`a=0` (b) `b=0` (c) `c=0` (d) nothing can be said about `a ,b ,c`.

Options

A. `a = 0`
B. `b = 0`
C. `c = 0`
D. nothing can be said about a, b, c.

Answer

1

Explanation

Suppose `a ne 0`. We rewrite f(x) as follows :
`f(x) = a{x^(2) + b/a x + c/a}`
`= {(x + (b)/(2a))^(2) + (4ac - b^(2))/(4a^(2))}`
`f(-(b)/(2a) + i) = a{(-(b)/(2a) + i + (b)/(2a))^(2) + (4ac - b^(2))/(4a^(2))}`
`= a{-1 + (4ac - b^(2))/(4a^(2))}`, which is a real number
This is against the hypothesis. Therefore, a = 0.