#### Answer

In the quadratic formula, $ x=\displaystyle \frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
we have a radicand, $ b^{2}-4ac $, which may be positive, zero, or negative.
The number of real solutions depends on its value, because a square root is taken from it.
The expression is called the discriminant, and when it is
- positive, there will be two distinct real solutions to the quadratic equation,
- zero, there will be one real solution (double real zero)
- negative, there will be no real solutions (the solutions are complex conjugates)

#### Work Step by Step

Given above.