February 25th, 2019, 09:42 AM  #1 
Newbie Joined: Aug 2015 From: USA Posts: 17 Thanks: 0  How is quadratic equation derived from Euclid?
Referring to Book 2 of Elements, given a segment AB, find a point C between A and B such that the lengths of segments satisfy AC > CB as well as the following proportionality relation: AC / AB = BC / AC Please show me how exactly this can be translated into the quadratic equation for x = AC / AB: x^2 + x  1 = 0 
February 25th, 2019, 12:08 PM  #2 
Newbie Joined: Feb 2019 From: United Kingdom Posts: 28 Thanks: 3 
What translation of the elements do you have? What proposition are you referring to?

February 25th, 2019, 04:49 PM  #3 
Newbie Joined: Aug 2015 From: USA Posts: 17 Thanks: 0 
Hi, please explain to me how the following can be translated to the quadratic equation below, if you can, thanks. Given a segment AB, find a point C between A and B such that the lengths of segments satisfy AC > CB as well as the following proportionality relation: AC / AB = BC / AC Please show me how exactly this can be translated into the quadratic equation for x = AC / AB: x^2 + x  1 = 0 
February 25th, 2019, 05:49 PM  #5  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,230 Thanks: 908 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Dan  
February 26th, 2019, 07:00 AM  #6 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
We can, without loss of generality, let AB= 1. Then x= AC/AB= AC. CB= AB AC= 1 x. $\displaystyle \frac{AC}{AB}= x= \frac{BC}{AC}= \frac{1 x}{x}$. Multiplying both sides by x, $\displaystyle x^2= 1 x$. Adding x to both sides and subtracting 1 from both sides, $\displaystyle x^2+ x 1= 0$. Last edited by Country Boy; February 26th, 2019 at 07:04 AM. 
February 26th, 2019, 06:40 PM  #7 
Newbie Joined: Aug 2015 From: USA Posts: 17 Thanks: 0 
That's wonderful how you did that Country Boy, first defining AB = 1. Might you recommend material to learn how to translate geometry to algebra like this? Thanks!

February 27th, 2019, 06:21 AM  #8 
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 551  It used to be studied under the name of "analytic geometry" in English. I do not think it is taught in the US public schools any longer, but look for that phrase in the title of a a book.

February 27th, 2019, 11:46 AM  #9 
Newbie Joined: Aug 2015 From: USA Posts: 17 Thanks: 0 
That's the one, thanks.

March 1st, 2019, 01:42 AM  #10 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,950 Thanks: 1141 Math Focus: Elementary mathematics and beyond  

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derived, equation, euclid, quadratic 
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