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January 22nd, 2012, 09:16 PM  #1 
Newbie Joined: Jan 2012 Posts: 10 Thanks: 0  Roots of the equation x2 –3x –5 = 0
Hi I have recently received some work to complete on a certain question that contains three problems. I can manage to complete the first of the problems although not the second, and without the second I cannot complete the third. The question written verbatim is as follows: Question 3 If ? and ? are the roots of the equation x^2–3x–5 = 0, find the values of the following expression, without actually finding the roots: a) 1/? + 1/? (express as a single fraction first) b) ?^2 + ?^2 (use: (a+b)^2=a^2+2ab+b^2 and rearrange) c) 1/?^2 + 1/?^2 (combine a) and b) above) The help would be so very much appreciated. I have been dwelling on these questions since I got them and nothing has seemed to come to me on how to do b) and c). Thanks. 
January 22nd, 2012, 09:44 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,209 Thanks: 517 Math Focus: Calculus/ODEs  Re: Roots of the equation x2 –3x –5 = 0 help need
If are the roots, then we may write: Thus, we have: and a) b) c) 
January 22nd, 2012, 09:46 PM  #3 
Newbie Joined: Jan 2012 Posts: 10 Thanks: 0  Re: Roots of the equation x2 –3x –5 = 0 help need
Thank you for the reply.

January 23rd, 2012, 02:26 PM  #4 
Senior Member Joined: Jul 2011 Posts: 245 Thanks: 0  Re: Roots of the equation x2 –3x –5 = 0
Darn you, Mark! I was hoping to get to typeset the beautiful Greek letters. :P

January 23rd, 2012, 04:18 PM  #5 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,209 Thanks: 517 Math Focus: Calculus/ODEs  Re: Roots of the equation x2 –3x –5 = 0
You could demonstrate how to find the desired quantities by using and ... 
January 23rd, 2012, 05:11 PM  #6 
Senior Member Joined: Jul 2011 Posts: 245 Thanks: 0  Re: Roots of the equation x2 –3x –5 = 0
I have something that is clever and ridiculous: Let there be two inverse functions: By virtue of inverse functions, we have: Thus, Jokes aside. . . I've kinda been pondering on precisely what you had in mind, Mark. I can't see anything worthwhile beside finding the value of alpha and beta directly from the quadratic formula. My imagination is being very bland. D: Due to time constraints, it seems I'll have to get back to you later. 
January 23rd, 2012, 06:15 PM  #7 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,209 Thanks: 517 Math Focus: Calculus/ODEs  Re: Roots of the equation x2 –3x –5 = 0
This is what I had in mind: (i) Since the discriminant of the given equation is not zero, the roots are distinct, and we may divide through by : (ii) Now the questions can be found as in my first post. 
January 24th, 2012, 01:20 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 20,089 Thanks: 1902 
1/? and 1/? are the roots of (1/x)²  3(1/x)  5 = 0, i.e., x² + (3/5)x  1/5 = 0, so 1/? + 1/? = 3/5, and by [color=#00AC00]MarkFL[/color]'s method for (b), 1/?² + 1/?² = (3/5)²  2(1/5) = 9/25 + 10/25 = 19/25. 

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