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I'm trying to solve the following equation for x:

$$ \sqrt{(x_1 - x)^2 + y_1^2} \;+\; \sqrt{(x_2 - x)^2 + y_2^2} \;+\; \sqrt{(x_3 - x)^2 + y_3^2} \;+\; \sqrt{(x_4 - x)^2 + y_4^2} \;=\; d $$

I understand that with two terms, it's possible to isolate one square root and square both sides to simplify the equation. However, with four terms, this method fails. Is there a way to approach this? Thanks!

Simon Mackenzie
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    Where does this equation come from? I suppose its origin is from the sum of distances of $4$ points from $(x,0)$ is $d$. – Integreek Oct 03 '24 at 04:30
  • Yes, we are looking for a point along a line such that the sum of the distance from 4 given points (x1,y1), (x2,y2), (x3,y3) and (x4,y4) is d. – Simon Mackenzie Oct 03 '24 at 04:38
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    This could be quite lengthy by hand, but maybe using a computer algebra system you can try the following. Subtract $d$ from both sides to arrange so that one side of the equation is zero. Then multiply both sides of the equation by a rationalizing factor appropriate for use with $5$ terms -- see the answers to this MSE question. This will clear the radicals. – Dave L. Renfro Oct 03 '24 at 04:55
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    Is this in general, or for a specific case (like in a contest math problem)? – Calvin Lin Oct 03 '24 at 07:48
  • @DaveL.Renfro Thanks this is close to what I was looking for. However even once the radicals are cleared you end up with a high order polynomial, and I don't know if you can find the roots easily. Not only that but most of the roots are not valid solutions to the initial equation. – Simon Mackenzie Oct 04 '24 at 12:07
  • @CalvinLin This is in general. – Simon Mackenzie Oct 04 '24 at 12:08
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    The reduction to a polynomial equation takes you to well-trodden territory at least. However, aside from the high degree, a significant problem is that the constants are not known, so even a qualitative analysis seems unlikely. For example, there are $9$ constants, so for example to roughly describe where the expression is greater than $d$ and where it's less than $d$ will probably involve specifying certain regions in $9$-dimensional space along with corresponding intervals of values for $x.$ – Dave L. Renfro Oct 04 '24 at 12:58

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