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Date: 4/11/96 at 13:37:19
From: Anonymous

I need to know why a quadratic equation cannot have one irrational
appreciated.

Thank you,

Kelli
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Date: 5/30/96 at 11:41:44
From: Doctor Gary

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.

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

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Associated Topics:
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