Solution of quadratic equation

Question

Can we have a quadratic equation $ax^2 +bx+c=0$ ,where $a,b,c$ are non-zero integers ($c$ is divisible by $a$ where $b$ isn’t) and one of its root is a fractional number and one is an integer? I couldn’t find any examples. I checked a few equations but getting both real roots only (none of it an integer).

Answer 1

Yes , it is possible .

Consider $f(x) = 2x^2-9x-18$

Clearly $2\mid18$ and $2\not\mid-9$ but $$f(x) = (x-6)(2x+3) $$

implying two roots at $x = 6$ and $x = -\dfrac 32.$