The discriminant is an important part of quadratic equations, which are written like this: $ax^2 + bx + c = 0$.

We calculate it using the formula $b^2 - 4ac$.

The discriminant helps us figure out how many solutions, or roots, the equation has, and what kind they are. Let’s break it down:

1. Positive Discriminant: If $b^2 - 4ac > 0$, there are two different real roots. For example, take the equation $x^2 - 5x + 6 = 0$. Here, the discriminant calculates to $(-5)^2 - 4(1)(6) = 25 - 24 = 1$. This means it has two solutions, which are $x = 2$ and $x = 3$.

2. Zero Discriminant: If $b^2 - 4ac = 0$, there is one real root. This is sometimes called a repeated root. For example, in the equation $x^2 - 4x + 4 = 0$, the discriminant is $(-4)^2 - 4(1)(4) = 16 - 16 = 0$. This tells us the solution is $x = 2$.

3. Negative Discriminant: If $b^2 - 4ac < 0$, this means there are no real roots, just complex roots. For example, in the equation $x^2 + 2x + 5 = 0$, the discriminant works out to $2^2 - 4(1)(5) = 4 - 20 = -16$. This shows us that the roots are complex and can’t be found on the regular number line.

Knowing about the discriminant helps us understand quadratic equations better and makes solving them easier!