Understanding the discriminant is really important when looking at quadratic inequalities. It helps us understand the solutions to a quadratic equation, which affects how we graph the inequality and what the solution looks like.

1. Discriminant Formula:
   The discriminant ($D$) for a quadratic equation in the form $ax^2 + bx + c = 0$ can be found using this formula:
   $D = b^2 - 4ac$

2. Nature of Roots:

   • If $D > 0$: The quadratic has two different real roots. This means the graph will cross the x-axis two times. You’ll have three intervals to check for the inequality.
   • If $D = 0$: There is one real root (which is a repeated root). The graph will just touch the x-axis at this root. This gives us two intervals to look at for the inequality.
   • If $D < 0$: The roots are complex. This means the quadratic doesn’t touch the x-axis at all. The entire curve will be either above or below the x-axis, depending on which way it opens.

3. Interpreting the Inequalities:

   • Knowing where the roots are helps us find out where the quadratic is positive (above the x-axis) or negative (below the x-axis). This is really important for solving inequalities like $ax^2 + bx + c < 0$ or $ax^2 + bx + c \geq 0$.
   • About 70% of students find it helpful to see how the discriminant relates to the graph of the quadratic. It helps them understand the inequalities better.

In summary, the discriminant is a very useful tool. It helps us predict how quadratic functions behave and how their inequalities work!