SOLUTION: Let f(x) = x^2 + kx and g(x)=x + k. The graphs of f and g intersect at two distinct points. Find the value(s) of k.

It means that the function f(x) - g(x) has two distinct real roots.


f(x) - g(x) = x^2 + kx - x - k = x^2 + (k-1)x - k.


The discriminant is  d = " b^2 - 4ac " = (k-1)^2 - 4*1*(-k) = k^2 - 2k + 1 + 4k = k^2 + 2k + 1 = (k+1)^2.



The function f(x) - g(x) has two distinct real roots if and only if the discriminant is positive.


It is true if and only if   k =/= -1.



ANSWER.  The graphs f and g intersect at two distinct points for any real number k different from -1.