Quadratic EquationHard

Question

Let $axx^{2} + bx + c$ is integer for all integral values of x , then which of the following must be true ?

Options

A.c is integer
B.$b = \frac{n}{2},n \in I$
C.$a = \frac{n}{2},n \in I$
D.$a,b$ both are integers

Solution

$f(x) = {ax}^{2} + bx + c \in I\forall x \in I$

$$\begin{matrix} & f(0) = c \in I \\ & f(1) = a + b + c \in I \\ & f( - 1) = a - b + c \in I \\ & \ \Rightarrow \ f(1) - f( - 1) = 2\text{ }b\ \Rightarrow \ 2\text{ }b \in I \\ & f(1) + f( - 1) = 2(a + c)\ \Rightarrow \ a = \frac{f(1) + f( - 1)}{2} - f(0) \\ & 2a \in I \\ & f(1) + f( - 1) = 2k_{1} + 1\text{~and~}f(1) - f( - 1) = 2k_{2} + 1 \\ & \text{~or~}\ b = \frac{2k_{2} + 1}{2},a = \frac{2k_{1} + 1}{2} - k = \frac{2k_{1} - 2k + 1}{2} \\ & f(1) + f( - 1) = 2k_{3}\text{~and~}f(1) - f( - 1) = 2k_{4},k_{i} \in I \\ & f(2) = 4a + 2\text{ }b + c \end{matrix}$$

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