If : $f(x) = 4x^2 - 4ax + b$ and $a^2-b$ is a perfect square of a rational number then among the following statements below which one is the right statement :
(a)If $a$ is odd , b is even , roots of $f(x) = 0$ are not integer
(b)If $a$ is even , b is odd , roots of $f(x) = 0$ are integer
(c)If $a$ is odd , b is odd , roots of $f(x) = 0$ are not integer
(d)If $a$ is even , b is even , roots of $f(x) = 0$ are not integer
I tried using Parity but I dont have a clear picture , can someone help me out
Let $a^2-b$ be perfect square of $k$ where $k\in Q$
Roots of $f(x)$ are $\displaystyle \alpha,\beta=\frac{a\pm\sqrt{a^2- b}}{2}=\frac{a\pm k}{2}$
IF $a,b$ are integers, then $k$ is also an integer
If $a$ is odd and $b$ is even , then $k$ is odd , therefore $a\pm k$ is even and hence roots of $f(x)$ are integers
Can you similarly check for other options?