What does the discriminant tell you?

When you look at the standard quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, the discriminant is just the piece underneath the square root. Its formula is $b^2 - 4ac$. We use it because the behavior of a square root dictates what kinds of answers we can get. Since you can only take the real square root of a positive number or zero, this small expression controls the entire outcome of the equation.

There are three possible outcomes when you calculate $b^2 - 4ac$. If the result is positive, you have a positive number under the square root, meaning you will add and subtract it to get two distinct real solutions. If the result is exactly zero, the square root of zero is zero, meaning the plus-or-minus part disappears and you get exactly one real solution. If the result is negative, you cannot take the square root of a negative number in the real number system, so you have zero real solutions (instead, you get two complex solutions).

The most common mistake is forgetting to put the quadratic equation into standard form before finding $a$, $b$, and $c$. The equation must look like $ax^2 + bx + c = 0$. If you have $x^2 + 5x = -6$, you cannot say $c$ is $-6$. You have to add 6 to both sides first. Another frequent trap is dropping negative signs when squaring $b$. Remember that $(-3)^2$ is positive 9, not negative 9.