When you solve for the roots of a quadratic equation, there are several possible outcomes.
There is a way to find out how many solutions there are before you even start using the formula. b^{2} – 4ac, called the discriminant, is the keystone species of our little quadratic ecosystem. Without it, the whole thing falls apart.
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} – 4ac is 0, then you have , so you have only one solution.
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} – 4ac = (0)^{2} – 4(1)(-3) = 12
So we have two real roots. Hah! Too easy.
How many roots does 2x^{2} + 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
b^{2} – 4ac = (8)^{2} – 4(2)(8) = 64 – 64 = 0
That means we have one real number root for this equation.
How many roots does 0.7731x^{2} – 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.
b^{2} – 4ac = (-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.
We may have been multitasking at the time. We're pretty busy, you know.
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