The Quadratic Formula is a helpful tool for solving quadratic equations.

A quadratic equation looks like this:

( ax^2 + bx + c = 0 )

The Quadratic Formula itself is:

( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )

Now, let's focus on an important part of this formula called the discriminant.

The discriminant is found by this formula:

( D = b^2 - 4ac )

Now, here’s what the discriminant tells us about the roots of the equation:

1. If ( D > 0 ): There are two different real roots.
   • For example, look at ( x^2 - 5x + 6 = 0 ).
2. If ( D = 0 ): There is one repeated real root.
   • An example is ( x^2 - 4x + 4 = 0 ).
3. If ( D < 0 ): There are no real roots, only complex roots.
   • A good example is ( x^2 + x + 1 = 0 ).

Using the discriminant helps us figure out the roots of the quadratic equation quickly, without having to solve the whole equation!