
Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion 
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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.) $\displaystyle N_{\lambda } =\frac{\displaystyle 1 + \lim_{s\rightarrow \infty} \left[2\left(1+e^{2s(14\lambda)}\right)+1\right]}{2}\lim_{s\rightarrow \infty} \left[2\left(1+e^{2s(14\lambda)}\right)+1\right]\cdot \left\lceil \frac{Re\{ \sqrt{14\lambda}\}+1}{Re \{ \sqrt{14\lambda}\}+2} \right\rceil.$ 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 $ ; $\displaystyle D=i\sqrt{5}$ ; $\displaystyle Re(D)=0$.
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?

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(14*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 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: $\displaystyle N_{\lambda } = \frac{\displaystyle 1 + \lim_{s\rightarrow \infty} \left[2\left(1+e^{2s(14\lambda)}\right)^{1}+1\right]}{2}\lim_{s\rightarrow \infty} \left[2\left(1+e^{2s(14\lambda)}\right)^{1}+1\right]\cdot \left\lceil \frac{Re\{ \sqrt{14\lambda}\}+1}{Re \{ \sqrt{14\lambda}\}+2} \right\rceil.$
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 .


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