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Solve for x

(x-b^2)/(a+c) + (x-c^2)/(a+b) + (x-a^2)/(b+c) = 4(a+b+c) How can I solve this equation for x ? Please show me the process and also the basic idea of solving this type of equation.

Note by Omar Sayeed Saimum
1 week, 4 days ago

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x=(a+b+c)^2 Md Hasib · 4 days, 7 hours ago

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What have you tried? Where did you get stuck?

Given that this is a linear equation in \(x\), it should be simple to shift terms around. (Of course, there may be other approaches.)

There is no guarantee that the answer you obtain will be "nice". Though, if you saw it in an olympiad, then it would likely be nice. Calvin Lin Staff · 1 week, 1 day ago

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@Calvin Lin My answer is not "nice". A big one. But it's an Olympiad question according to my friend ( He gave me this problem to solve). So..... Omar Sayeed Saimum · 1 week, 1 day ago

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@Omar Sayeed Saimum Can you state your answer? Ideally, make it as simplified as possible. Calvin Lin Staff · 1 week, 1 day ago

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@Calvin Lin X= (4(a+b+c)^2(ab+bc+ca) + (a+b+c)(a^3+b^3+c^3))/((a+b+c)^2 + (ab+bc+ca)) Omar Sayeed Saimum · 1 week, 1 day ago

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@Omar Sayeed Saimum Hm, that doesn't seem quite right to me. Can you lay out all of your steps?

  1. The terms should be degree 2, but your first term has degree 4.
  2. In the 2nd term, you are multiplying by (a+b+c) and then dividing by (a+b+c)^2, so that could be simplified.
Calvin Lin Staff · 1 week, 1 day ago

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@Calvin Lin Please look at that again. the total term is divided by ((a+b+c)^2 + (ab+bc+ca)). Omar Sayeed Saimum · 1 week, 1 day ago

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@Omar Sayeed Saimum Ah ic. Missed the brackets because none of the terms made sense to me.

I agree with the denominator being \( (a+b)(b+c) + (b+c)(c+a) + (c+a)(a+b) \). It is more natural to write it as \( a^2 + b^2 + c^3 + 3ab + 3bc + 3ca \).

I'm not sure how you arrived at the numerator though. To me, it should be \( b^2(b+c)(b+a) + c^2 (c+a)(c+b) + a^2(a+b)(a+c) + 4(a+b+c)(a+b)(b+c)(c+a) \).

As always, if you lay out your steps, someone would be able to see what you did and provide feedback from there. But otherwise, we can only guess at what you're doing. Calvin Lin Staff · 1 week, 1 day ago

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@Calvin Lin Yeah, the denominator is a^2 + b^2 + c^2 + 3(ab+bc+ca) I've just separated the term using (a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab+bc+ca) Anyway, if you prefer to lay out all my steps, then I'll do so. Oh, I think the answer is (a+b+c)^2. I was using some terms in the equation randomly. Then suddenly I noticed that if we use x = (a+b+c)^2, the equation becomes true. Omar Sayeed Saimum · 1 week, 1 day ago

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@Omar Sayeed Saimum Yes, that's the answer. If you had the right numerator, you would be able to divide it out to arrive at that value.

Other than making a lucky observation about the answer, how else can we appraoch this problem? Calvin Lin Staff · 1 week, 1 day ago

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@Calvin Lin That's what I'm asking to you. Still I'm trying to get this. But can't find anyway. Wil you kindly show me the process if you get that? Then I'll be relieved from this headache.😊 Omar Sayeed Saimum · 1 week, 1 day ago

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@Omar Sayeed Saimum tl;dr I would have expanded everything to get the answer because it's an obvious step that works. There are some observations that could be made that eventually arrive at the answer, but this might take longer than brute force expansion.

Here are some observations:

  1. The LHS is linear in \( x \).
  2. The coefficient of \(x\) is pretty ugly, like \( \frac{ a^2+b^2+c^2 + 3ab + 3bc + 3ac } { (a+b)(b+c)(c+a) } \).
  3. If we were to change the RHS to almost any other term, there is no nice solution that we can find. This suggests that the answer is "specially determined".
  4. (I might be wrong here) There isn't a nice way to simplify the LHS. At the most, a substitution of \( s = a+b+c \) might help, but I don't see how else to proceed. At this point in time, I would be tempted to brute force the calculations, which was what I did initially. We get the nice answer of \( (a+b+c)^2 \), which suggests something else might be going on.
  5. Looking at the problem again, observe that if \( a+b = 0 \) then we must have \( x - c^2 \equiv 0 \pmod{a+b} \). If we tried to solve the system of equivalence equations, namely \( x = c^2 + k_c (a+b) = a^2 + k_a (b+c) = b^2 + k_b( c+a) \), we might eventually arrive at \( x \equiv (a+b+c)^2 \pmod{ (a+b)(b+c)(c+a) } \). (Note: I say might because there isn't a standard approach to solving equivalence relations in variables.) We thus then plug in \( x = (a+b+c)^2 + K (a+b)(b+c)(c+a) \) into the equation, and observe that we get the RHS when \( K = 0 \).
Calvin Lin Staff · 6 days, 5 hours ago

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@Calvin Lin Thanks. But I didn't understand the 3rd line of the 5th step. x = (a+b+c)^2 (mod(a+b)(b+c)(c+a)) How? Omar Sayeed Saimum · 6 days, 4 hours ago

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@Omar Sayeed Saimum

Note: I say might because there isn't a standard approach to solving equivalence relations in variables.

It is clear that \( x = (a+b+c)^2 \) satisfies \( x = c^2 + k_c (a+b) = a^2 + k_a (b+c) = b^2 + k_b( c+a) \), but it's not clear how we can easily backtrack. Calvin Lin Staff · 6 days, 3 hours ago

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Calvin Lin plz help me.

Omar Sayeed Saimum · 1 week, 3 days ago

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@Omar Sayeed Saimum Instead of asking people for help, tell us where you're stuck because we don't know what issues you're facing. Pi Han Goh · 1 week, 3 days ago

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@Pi Han Goh I didn't solve this kind of equation before. I'm trying. But couldn't find any way. So please help me to solve this. Omar Sayeed Saimum · 1 week, 3 days ago

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