The discriminant, which is shown by the formula $b^2 - 4ac$, helps us learn about the solutions, or roots, of a quadratic equation in the form $ax^2 + bx + c = 0$.

Here’s what the discriminant tells us:

1. Real and Different Roots: If the discriminant is positive (more than 0), the equation has two different real roots. This means we can break it down into two simpler parts, like this:

   $x^2 - 5x + 6 = 0 \quad \text{(discriminant = } 1)$
   Factors: $(x-2)(x-3)$.

2. One Repeated Root: If the discriminant is zero (exactly 0), there is one root that repeats. In this case, we can factor it like this:

   $x^2 - 4x + 4 = 0 \quad \text{(discriminant = } 0)$
   Factors: $(x-2)^2$.

3. Complex Roots: If the discriminant is negative (less than 0), the roots are complex, which means we can't break it down into simple real factors. For example:

   $x^2 + 4x + 8 = 0 \quad \text{(discriminant = } -16)$
   Factors: These aren't real numbers, and we show them as $x = -2 \pm 2i$.

In short, the discriminant helps us see if a quadratic can be simplified into real numbers or if it will lead us to more complex solutions.