Q.The polynomial $x^3-kx^2+9x$ has three real zeros if and only if, $k$ satisfies

Restricting all the way to integer coefficients (or, equivalently after clearing denominators, rational coefficients) sharpens the quadratic discriminant test into an exact rationality criterion. For $ax^2+bx+c=0$ with integer $a,b,c$, $\Delta=b^2-4ac$ is automatically an integer, and

So an integer-coefficient quadratic has rational roots exactly when its discriminant is a non-negative perfect square; otherwise (with $\Delta>0$ but not a perfect square) the roots are real but irrational; with $\Delta<0$ the roots are non-real (imaginary); with $\Delta=0$ the (repeated) root is real and, in fact, rational.