Ms. Harmony's Website of Mathematical Fabulousness
Home
For Students
The Need-To-Knows
Being a Good Student Means...
Integers
Fractions
Area & Perimeter
Solving Equations
>
Intro - Solving Equations
Solving Equations Lessons
Three Types of Answers to Equations
Quadratics
Introduction to Quadratics
Why are Parabolas Useful/Important
Imaginary Roots
Constructions
Isometry
Vocab
Contact
Imaginary Roots
"
Imaginary" or complex roots in math are complex numbers other than real numbers which solve an equation. For example, the equation
x² + 1 = 0
has two complex roots, x = i and x = -i, where i is the symbol designating the square root of -1.
It's not always easy to say whether an equation has imaginary roots. It's easier withpolynomial equations, because the Fundamental Theorem of Algebra says that an nth-degree polynomial always has n (not necessarily distinct) roots in the complex field. The field of complex numbers includes reals, so the the number of real roots and "imaginary" roots of an nth degree polynomial must together be n. So if you rule out all real roots, the others must be complex.
For example consider the 3rd degree polynomial x³ - 1. We know that the equation
x³ - 1 = 0
has one real root at x = 1. Factor that out:
x³ - 1 = (x - 1)(x² + x + 1)
And you'll find that the remaining factor x² + x + 1 has no real roots. By the Fundamental Theorem of Algebra, there are two roots remaining so they must be complex. In this case, you can use the quadratic formula to find them:
x = [-1 ± √(1 - 4)] / 2
Thus the remaining two roots are the complex conjugates
x = -1/2 + (√3/2)i
x = -1/2 - (√3/2)i