Students can use something called the discriminant, which is shown as ( D = b^2 - 4ac ), to figure out what kind of roots a quadratic function has. Quadratic functions look like this: ( ax^2 + bx + c = 0 ).

What the Discriminant Tells Us:

1. Positive Discriminant (( D > 0 )):

   • This means there are two different real roots.
   • For example, in the equation ( x^2 - 5x + 6 = 0 ):
     • Here, ( D = (-5)^2 - 4(1)(6) = 25 - 24 = 1 ), which is greater than 0.

2. Zero Discriminant (( D = 0 )):

   • This means there is one real root that repeats (we call it a double root).
   • For example, in the equation ( x^2 - 4x + 4 = 0 ):
     • Here, ( D = (-4)^2 - 4(1)(4) = 16 - 16 = 0 ).

3. Negative Discriminant (( D < 0 )):

   • This means there are no real roots; instead, the roots are complex (which we often call imaginary).
   • For example, in the equation ( x^2 + x + 1 = 0 ):
     • Here, ( D = 1^2 - 4(1)(1) = 1 - 4 = -3 ), which is less than 0.

In Summary:

By calculating the discriminant, students can easily find out how many roots a quadratic equation has and what type they are.