 My Math Forum Need help to verify the number of solutions

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 October 29th, 2019, 09:10 AM #1 Senior Member   Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math Need help to verify the number of solutions Given equation $\displaystyle x^2 +x +\lambda =0 \; ,x\in \mathbb{R}$. Verify N - the number of solutions, using any software (matlab, calculators... etc.) $\displaystyle N_{\lambda } =\frac1 + \lim_{s\rightarrow \infty} \left[-2\left(1+e^{-2s(-1-4\lambda)}\right)+1\right]}{2}\lim_{s\rightarrow \infty} \left[-2\left(1+e^{-2s(-1-4\lambda)}\right)+1\right]\cdot \left\lceil \frac{Re\{ \sqrt{-1-4\lambda}\}+1}{Re \{ \sqrt{-1-4\lambda}\}+2} \right\rceil$ Re - real part of imaginary number (if it exists). Last edited by skipjack; October 30th, 2019 at 02:43 PM. October 29th, 2019, 09:29 AM #2 Member   Joined: May 2013 Posts: 57 Thanks: 5 x is a real number. not an imaginary or complex number. x^2 +x +L = 0 x = (1 +/- sqrt(1 -4*L))/2 Last edited by phillip1882; October 29th, 2019 at 10:00 AM. October 29th, 2019, 10:58 AM #3 Senior Member   Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math For example $\displaystyle \lambda =1$ ; $\displaystyle D=i\sqrt{5}$ ; $\displaystyle Re(D)=0$. Thanks from topsquark Last edited by idontknow; October 29th, 2019 at 11:18 AM. October 29th, 2019, 01:21 PM #4 Global Moderator   Joined: May 2007 Posts: 6,852 Thanks: 743 Why is this question repeated? Thanks from topsquark October 30th, 2019, 12:48 PM #5 Member   Joined: May 2013 Posts: 57 Thanks: 5 it specifically states in the first sentence, x is a real number also, if L = 1; (1 +/- sqrt(1-4*1))/2 (1 +/- sqrt(-3))/2 1+/- i*sqrt(3))/2 x = 1/2 +i*sqrt(3)/2 or x = 1/2 -i*sqrt(3)/2 so the real part is 1/2. and will be 1/2 for any L value > 1/4 Thanks from idontknow Last edited by phillip1882; October 30th, 2019 at 12:52 PM. October 30th, 2019, 01:10 PM #6 Senior Member   Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math I fixed the exponents now: $\displaystyle N_{\lambda } = \frac1 + \lim_{s\rightarrow \infty} \left[-2\left(1+e^{-2s(-1-4\lambda)}\right)^{-1}+1\right]}{2}\lim_{s\rightarrow \infty} \left[-2\left(1+e^{-2s(-1-4\lambda)}\right)^{-1}+1\right]\cdot \left\lceil \frac{Re\{ \sqrt{-1-4\lambda}\}+1}{Re \{ \sqrt{-1-4\lambda}\}+2} \right\rceil$ Last edited by skipjack; October 30th, 2019 at 02:47 PM. November 2nd, 2019, 07:54 AM #7 Senior Member   Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math Just plug the value of $\displaystyle \lambda$ into $\displaystyle N(\lambda )$ and it gives the number of solutions . Tags hellp, number, relation, solutions, verify Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post shindigg Calculus 7 March 4th, 2017 09:52 PM Chikis Algebra 15 December 29th, 2015 07:47 AM uniquegel Algebra 4 September 8th, 2014 05:18 PM Rakshasa Applied Math 7 May 7th, 2012 02:37 PM sivela Calculus 1 April 13th, 2010 12:28 PM

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