How Many Roots? at a Glance

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. 

Don't worry; there's an easy way to find out how many solutions there are before you even start using the formula. Just take a peek at the b² – 4ac part of the quadratic formula. That little chunk is called the discriminant, and it's the keystone species of our little quadratic ecosystem. Without it, the whole thing falls apart.

• If b² – 4ac is positive, then there are two real number solutions. 
    
• If b² – 4ac = 0, then there's only one real number solution. 
    
• If b² – 4ac 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's negative, you have , which gives two complex results. And if b² – 4ac is 0, then you have , so you have only one solution.

Sample Problem

How many roots does x² – 3 = 0 have? 

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

b² – 4ac = (0)² – 4(1)(-3) = 12 

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

Okay, How About This?

How many roots does 2x² + 8x + 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

b² – 4ac = (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 0.7731x² – 2.3812x + 4.1111 = 0 have?

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

b² – 4ac = (-2.3812)² – 4(0.7731)(4.1111) ≈ 5.6701 – 12.7132 = -7.0431

That's negative, so 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.