Rationalizing Denominator: What Went Wrong in Multiplying by Conjugate?

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Homework Help Overview

The problem involves rationalizing the denominator of the expression \(\frac{x^2-9}{\sqrt{3-x}}\). Participants are discussing the correct approach to simplify this expression, particularly focusing on the use of conjugates in the rationalization process.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the necessity of multiplying by the conjugate and question the correct form of the conjugate in this context. There is a discussion about whether to multiply by \(\sqrt{3+x}\) or \(\sqrt{3-x}\) and the implications of treating the square root as a single entity.

Discussion Status

The discussion is ongoing, with participants providing insights into the rationalization process and questioning assumptions about when to use conjugates. Some guidance has been offered regarding the manipulation of the radical, but no consensus has been reached on the best approach.

Contextual Notes

Participants are navigating the rules of rationalization and the specific characteristics of the expression, including the nature of the square root in the denominator. There is an emphasis on understanding the role of conjugates in this scenario.

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Homework Statement


Rationalize the denominator and simplify:

\frac{x^2-9}{\sqrt{3-x}}


Homework Equations


None.


The Attempt at a Solution


The answer to the question is -(x+3)\sqrt{3-x}, but this is what I'am getting:
http://img222.imageshack.us/img222/9554/radicalio9.jpg
Can someone show me where I went wrong?
 
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Well, when you rationalize a problem. You want to get rid of the radical. In order for the radical to go away, you have to manipulate it so that the power is 1.

You multiplied the both numerator and denominator by \sqrt{3+x} when you should have multiplied both by \sqrt{3-x}
 
But I thought that when you rationalize, you're supposed to multiply the numerator and denominator by the conjugate.
 
Since we have a square root, it's like a grouping symbol, we have to take the entire piece as one single thing. So we we let \sqrt{3+x} = a, then there's really only one thing in the denominator. We only multiply by the conjugate when the identity p^2-q^2 = (p+q)(p-q) is useful to us, in this case not. This time it was easier just to get rid of the square root.

Or another way to think of it, you have a+0 in the denominator, you have to multiply by its conjugate, a-0 = a.
 
temaire said:
But I thought that when you rationalize, you're supposed to multiply the numerator and denominator by the conjugate.


Yes, and what is the conjugate here? The conjugate of a+ b\sqrt{c} is a- b\sqrt{c}. What are a, b, and c here?
 

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