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Sunday, 01 May 2011 17:14

Complex Numbers and Quadratic Polynomials

By  Gideon
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Recall that a polynomial of degree two is called a Quadratic Polynomial.

A general quadratic polynomial (over the real numbers) is of the form:

\[ az^{2} + bz + c = 0. \]

The formula for solving for the roots of this equation is:

\[ z = \frac{-b \pm \sqrt{b^{2} -4ac}}{2a} .\]

The term: \[ b^{2} -4ac, \]is called the Discriminant of the polynomial.

If the discriminant is less than zero, then the polynomial has no real roots. (There are no real numbers that square to negative numbers.)

If we call the discriminant \[ D = d^{2,} \]then when D < 0, d will be imaginary, and the roots will be of the form:

\[ z = \frac{-b}{2a} \pm \frac{\sqrt{d^{2}}}{2a} \]

\[ = \frac{-b}{2a} \pm i\frac{|d|}{2a} .\]

In other words, when D < 0, the roots will have the following properties:

  • there will be two distinct roots;
  • the roots will be complex numbers:

\[ z_+ = x + iy = \frac{b}{2a} + i\frac{|d|}{2a}, \]

and,

\[ z_- = x - iy = \frac{b}{2a} - i \frac{ |d| }{2a}; \]

  • the real parts of the roots will be equal:

\[ Re(z_+) = Re(z_-) = x; \]

  • the imaginary parts of the roots will be additive inverses of each other:

\[ Im(z_+) = y, \]

\[ Im(z_-) = -y .\]

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