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.

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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 aramuseum.org massage room, next to a pile of Algebra textbooks. In case you were wondering.

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