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# Quadratic Formula and Functions

# How Many Roots?

When you solve for the roots of a quadratic equation, there are several possible outcomes.

- You can have two real number solutions. If you set
*x*equal to either solution, the result with be zero both times.

- There can be just one real number solution.

- The equation can have two complex number solutions. There are no real number solutions.

There is a way to find out how many solutions there are before you even start using the formula. *b*^{2} – 4*ac*, called the **discriminant**, is the keystone species of our little quadratic ecosystem. Without it, the whole thing falls apart.

- If
*b*^{2}– 4*ac*is positive, then there are two real number solutions.

- If
*b*^{2}– 4*ac*= 0, then there is only one real number solution.

- If
*b*^{2}– 4*ac*is negative, then there are two complex number solutions.

This all comes directly from the quadratic formula. If the discriminant is positive, then you have , which leads to two real number answers. If it is negative, you have , which gives two complex results. And if *b*^{2} – 4*ac* is 0, then you have , so you have only one solution.

### Sample Problem

How many roots does *x*^{2} – 3 = 0 have?

To use the discriminant, we first note that *a* = 1, *b* = 0, and *c* = -3.

*b*^{2} – 4*ac* = (0)^{2} – 4(1)(-3) = 12

So we have two real roots. Hah! Too easy.

### Okay, How About This?

How many roots does 2*x*^{2} + 8*x* + 8 = 0 have?

Hey now, stop it with that lip, Subheading. Why not just say "Sample Problem" like you usually do? Anyway, the discriminant for this equation is

*b*^{2} – 4*ac* = (8)^{2} – 4(2)(8) = 64 – 64 = 0

That means we have one real number root for this equation.

### How Do You Like This One, Then?

How many roots does 0.7731*x*^{2} – 2.3812*x* + 4.1111 = 0 have?

Now that's just being mean—but we can still do it. Just let us find our calculator real quick.

*b*^{2} – 4*ac* = (-2.3812)^{2} – 4(0.7731)(4.1111) = 5.6701 – 12.7132 = -7.0431

There are two complex roots for this equation. Also, the calculator was in the Shmoop massage room, next to a pile of Algebra textbooks. In case you were wondering.

### What Was It Doing There?

We may have been multitasking at the time. We're pretty busy, you know.