Q.If $k$ is real, discuss the nature of the roots of the polynomial equation $2x^2+kx+k=0$, in terms of $k$.

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.

Reading the nature of roots off a parameter, without solving. A parametrised quadratic like $ax^2+kx+k=0$ has its nature governed entirely by the sign of $\Delta(k)=k^2-4ak$ as $k$ varies — factoring or sign-analysing $\Delta(k)$ splits the real line of $k$-values into ranges giving real-distinct, real-equal, or non-real roots, all without ever writing down the roots themselves. The same idea underlies proving a quadratic-in-disguise always has rational roots, purely by showing its discriminant reduces to a perfect square symbolically (e.g. $4(q-r)^2$ for rational $q,r$) — again with no need to actually solve for the roots.