The discriminant in a quadratic equation is a key tool we use to learn about the solutions, or roots, of that equation. It is calculated using the formula ( b^2 - 4ac ), where ( a ), ( b ), and ( c ) are the numbers from the equation.

Let’s break it down based on the value of the discriminant:

1. Positive Discriminant ((b^2 - 4ac > 0)):

   • This means there are two different real roots.
   • For example, take the equation ( x^2 - 3x + 2 = 0 ).
   • Here, the discriminant is calculated as: [ (-3)^2 - 4(1)(2) = 1 ]
   • Since the discriminant is positive, this tells us there are two different solutions.

2. Zero Discriminant ((b^2 - 4ac = 0)):

   • This indicates there is one real root, also known as a repeated root.
   • For instance, in the equation ( x^2 - 4x + 4 = 0 ), the discriminant is: [ 0 ]
   • This means there is just one unique solution, which is ( x = 2 ).

3. Negative Discriminant ((b^2 - 4ac < 0)):

   • This means there are no real roots at all, only complex roots.
   • For example, consider the equation ( x^2 + x + 1 = 0 ).
   • In this case, the discriminant is calculated as: [ 1 - 4 = -3 ]
   • Since the discriminant is negative, this indicates that there are two complex roots.

In summary, the discriminant is super helpful! It tells us how many solutions a quadratic equation has and what type they are.