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

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.