Originally Posted by
HallsofIvy Recall that the solutions to the quadratic equation $\displaystyle ax^2+ bx+ c= 0$ are, by the quadratic formula, $\displaystyle \frac{-b\pm\sqrt{b^2- 4ac}}{2a}$. Whether there are two real roots, one double root, or two complex roots depends upon whether the number in the square root, $\displaystyle b^2- 4ac$ is positive, zero, or negative.
That number, $\displaystyle b^2- 4ac$ is the "discriminant".
As row said, in your case, the discriminant is -71. That tells us that the equation $\displaystyle 3x^2+ 5x+ 8= 0$ has two complex roots.
It als tells us that the graph of $\displaystyle y= 3x^2+ 5x+ 8$ does not cross the x-axis but is entirely in the upper half-plane.