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Topic: Enrichment
Replies: 38   Last Post: Aug 24, 2012 1:33 PM

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Dave L. Renfro

Posts: 4,518
Registered: 12/3/04
Re: Enrichment
Posted: Aug 22, 2012 11:06 AM
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Dave L. Renfro wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=7871436

>> Mike's function, f(x) = (x-1)(x-2) / (x+1)(x+2),
>> is a quadratic divided by a quadratic.


[snip]

>> We begin by rewriting the equation in implicit form
>> as a quadratic expression in the variable x:
>>
>> x^2y + 3xy + 2y = x^2 - 3x + 2
>>
>> (y-1)x^2 + 3(y+1)x + 2(x-1) = 0


Kirby Urner wrote:

http://mathforum.org/kb/message.jspa?messageID=7872321

> I got lost with the words "We begin by rewriting
> the equation..."
>
> Which equation again? Not Mike's. Not Darboux's.
>
> I wasn't able to find an equation in x and y only,
> that's being rewritten.


I was putting Mike's equation in implicit form,
after replacing f(x) by y (since we're working in
the standard (x,y) coordinate plane).

y = (x-1)(x-2) / (x+1)(x+2)

y = (x^2 - 3x + 2) / (x^2 + 3x + 2)

y(x^2 + 3x + 2) = x^2 - 3x + 2

yx^2 + 3xy + 2y = x^2 - 3x + 2

yx^2 + 3xy + 2y - x^2 + 3x - 2 = 0

(yx^2 - x^2) + (3xy + 3x) + (2y - 2) = 0

(y-1)x^2 + 3(y+1)x + 2(y-1) = 0

I see now that there is a typo in my last equation
quoted in your post, however. I wrote "2(x-1)",
but it should have been "2(y-1)".

Dave L. Renfro



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