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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

Lalit Tolani
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RDT
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1 Answers1

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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?

Lalit Tolani
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