The Discriminant, which we write as ( D = b^2 - 4ac ), is very important when we want to understand the solutions of a quadratic equation. A quadratic equation looks like this: ( ax^2 + bx + c = 0 ).

Let's break down what the Discriminant tells us:

1. If ( D > 0 ): This means there are two different real solutions.

   • For instance, take the equation ( x^2 - 5x + 6 = 0 ).
   • Here, ( D = (-5)^2 - 4(1)(6) = 25 - 24 = 1 ), which is greater than 0. So, there are two solutions!

2. If ( D = 0 ): There is exactly one real solution, which is like a repeated answer.

   • For example, in the equation ( x^2 - 4x + 4 = 0 ),
   • We calculate ( D = (-4)^2 - 4(1)(4) = 0 ). This means there is one solution.

3. If ( D < 0 ): There are no real solutions here. Instead, we have complex solutions.

   • Take the equation ( x^2 + x + 1 = 0 ).
   • We find ( D = (1)^2 - 4(1)(1) = 1 - 4 = -3 ), which is less than 0. So, there are no real solutions.

In short, the Discriminant helps us figure out what type of solutions we can find!