real solutions: $c \in (- \infty, -2] \cup [2, \infty)$
Work Step by Step
The equation $x^2 +cx+1$ has real roots if the discriminant $(c^2 - 4) \ge 0$ from the general law for solving a polynomial of second degree. Thus, we need to have: $|c| \ge 2$ which can be written as $c \in (- \infty, -2] \cup [2, \infty)$ Thus, any value in these intervals gives us a real solution. But any value of $c$ in the interval $(-2,2)$ gives us an imaginary (non-real) solution.