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 December 17th, 2012, 11:29 PM #1 Senior Member   Joined: Dec 2012 Posts: 148 Thanks: 0 Find x and y. $x^2 +y= 10 \\ x + y^2 = 4$ I know answer is (3;1) but I am looking for another root(s). I got this $\fbox{ x= \frac{ 6 + a + a^2 }{2a} \\ y = \frac{ 6 + a - a^2 }{2a} }$
 December 17th, 2012, 11:45 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Find x and y. I would use the first equation, and substitute into the second, to get the quartic in $x$: $x+$$10-x^2$$^2=4$ $x^4-20x^2+x+96=0$ $(x-3)$$x^3+3x^2-11x-32$$=0$ and then use a numeric root finding technique to approximate the other 3 irrational roots: $x\approx-3.5709287407546415449,\,-2.7216512006427414436,\,3.2925799413973829885$
 December 18th, 2012, 04:12 AM #3 Senior Member   Joined: Dec 2012 Posts: 148 Thanks: 0 Re: Find x and y. Only one of these irrational roots may be correct. I analyzed my incomplete solution $4 \ll a \ll 5$ . a is near to 4
 December 18th, 2012, 04:17 AM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Find x and y. They are all correct. You have made an error with your incomplete solution.
December 18th, 2012, 04:51 AM   #5
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Re: Find x and y.

Quote:
 Originally Posted by MarkFL They are all correct. You have made an error with your incomplete solution.

 December 18th, 2012, 05:09 AM #6 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Find x and y. I have already shown the numerically approximated roots. I'm not going to manually grind out the 3 irrational roots using the Newton-Raphson method, since that is a technique from differential calculus. If you feel incomplete with these roots as given, then try the quartic formula.
December 18th, 2012, 05:33 AM   #7
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Re: Find x and y.

Quote:
 Originally Posted by MarkFL I have already shown the numerically approximated roots. I'm not going to manually grind out the 3 irrational roots using the Newton-Raphson method, since that is a technique from differential calculus. If you feel incomplete with these roots as given, then try the quartic formula.
OK, I will try, after watching american horror story:asylum

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