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I have this equation: $$0=\frac{8}{\sqrt{30^2-w^2}}+\frac{8}{\sqrt{20^2-w^2}}-1$$

But I need to express it as a polynomial equation, or an equivalent equation that is also polynomial, I have tried different transformations but I honestly can't find how to do it.

[EDIT]: Im solving this problem of the book of Burden Problem of Burden, after the Pythagorean theorem andsome Thales's, result this equation. I need to solve it with Mullers Method, i already solved it, but my teacher says that this type of equations should not be expressed in this way, it should be changed to a polynomial equation or a equivalent one that does not have roots.

[EDIT 2]: Following the comment of Gerry, the result it takes something like this $$1^2=\left(\frac{8}{\sqrt{30^2-w^2}}+\frac{8}{\sqrt{20^2-w^2}}\right)^2$$

Result something like this, but i dont know how proceed Ecuation square

paez49
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  • Welcome to MSE. In this case, and as a general rule when asking a question, it would be helpful if you showed us precisely how you have tried to solve the problem. This will help users better answer your question, and will let them know that you are interested in learning, and have put effort into coming up with a solution. – Christian E. Ramirez Sep 24 '22 at 05:46
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    Okay i will post more information – paez49 Sep 24 '22 at 05:52
  • Does this answer your question? Crossed Ladders Problem – Parcly Taxel Sep 24 '22 at 06:04
  • Add $1$ to both sides. Square both sides. Isolate the square root, and square both sides again. Clear fractions. – Gerry Myerson Sep 24 '22 at 06:07
  • @ParclyTaxel No because the problem is not solve literally the problem, the problem is make this equation different – paez49 Sep 24 '22 at 06:11
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    @GerryMyerson my english level is low, i dont understand a specific math terms like isolate the square root, can you show me how make it in maths terms – paez49 Sep 24 '22 at 06:14
  • "Isolate the square root" means rearrange the equation so the square root is all by itself on one side of the equation. – Gerry Myerson Sep 24 '22 at 06:28
  • @GerryMyerson ok i understand, but in the edit 2 how can i isolate the roots, if the whole sum is squared – paez49 Sep 24 '22 at 07:19
  • @JuanDavidPáezSánchez First, you have to square the term on the left hand side of the equation shown in your edit (2). That will give you an equation, the left hand side of which contains a term with root sign. Now, you can bring this term with the root sign to a one side and the other terms (including 1) to the other side. Finally, square both sides of this equation to obtain an equation free of radicals. – YNK Sep 24 '22 at 09:25
  • Maybe this form helps after common denominators? – Тyma Gaidash Sep 24 '22 at 11:55
  • OK, Juan, so now you have something of the form, $(a+b)^2$. Surely you have come across the formula, $(a+b)^2=a^2+2ab+b^2$? – Gerry Myerson Sep 24 '22 at 22:32
  • Ok thanks Gerry and YNK, I thought the solution to the problem was much less cumbersome than described, I'll try that another day but I'm still getting feedback on how to do it. – paez49 Sep 25 '22 at 01:10
  • @TymaGaidash is a good transformation, but still have roots :// – paez49 Sep 25 '22 at 01:13
  • It appears to me that when all is said and done, you'll wind up with a polynomial equation of degree four in the variable $w^2$. – Gerry Myerson Sep 27 '22 at 06:08

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