TITLE: Showing that roots of a quadratic polynomial are of opposite signs.
QUESTION [9 upvotes]: Prove that, for all values of $k$, the roots of the quadratic polynomial $x^2 - (2 + k) x - 3$ are real. Show further that the roots are of opposite signs.

For the first part I was able to demonstrate such by using the discriminant of the quadratic, then using the discriminant of the discriminant. 
For the second part I was not able to demonstrate such.

REPLY [22 votes]: In the quadratic $P(x) = x^2 - (2 + k)x - 3 = 0$, the coefficient of $x^2$ is positive. Thus, $P(X)$ and $P(-X)$ is positive for some large enough $X>0$.
But $P(0) = -3$ is negative. So between $-X$ and $0$ there is a $x_1$ where $P(x_1) = 0$ and between $0$ and $X$ there is a $x_2$ where $P(x_2) = 0$: the polynomial has real roots and they are of opposite signs.