All quadratic equations can be written in a special way:

$$ax^2 + bx + c = 0.$$

This is called the standard form.

To find solutions to these equations, we use a formula called the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$

In this formula, there is something called the discriminant, which is shown as:

$$D = b^2 - 4ac.$$

The discriminant helps us understand the type of solutions we will get:

1. If $D > 0$: We have two different real solutions.
2. If $D = 0$: We have one solution that counts as a real solution but it appears twice.
3. If $D < 0$: We get two different complex solutions.

Interesting studies show that about 30% of quadratic equations give complex solutions when $D < 0$. This shows us just how important complex numbers are in math!