Quadratic Roots

Date: 4/11/96 at 13:37:19
From: Anonymous
Subject: Algebra: quadratic equations

I need to know why a quadratic equation cannot have one irrational 
or one imaginary root. Your help in answering this would be 
appreciated.

Thank you,

Kelli

Date: 5/30/96 at 11:41:44
From: Doctor Gary
Subject: Re: Algebra: quadratic equations

Dear Kelli;

 Here are two explanations.

 1. Do you know how to "build" a quadratic equation?   Once you 
learn how, you'll be able to understand why a quadratic equation 
can't have just one irrational or imaginary root.

 Quadratic equations take the form:

  ax^2 + bx + c  =  0, in which a, b, and c are integers.

 Let's suppose we wanted to "create" a quadratic equation whose 
roots were "r" and "s".   We would start out with:

  (x - r)(x - s)  =  0

 When we multiply the two terms on the left side of this equation, 
we see that:

  x^2 - (r+s)x + rs  =  0

 Look at the middle or final terms.  If either r or s is imaginary 
(or irrational) and the other one isn't, then the sum (r+s) and/or 
the product rs won't be rational.

  2. The quadratic formula, which represents the general 
solution to quadratic equations provides another explanation.  
If sqrt(b^2 - 4ac) is irrational, then there will be two 
irrational roots.  If sqrt(b^2 - 4ac) is imaginary, then there 
will be two imaginary (or "complex") roots.    

-Doctor Gary,  The Math Forum