Quadratic RootsDate: 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 |
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