What are the four roots of this challenging equation?

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Discussion Overview

The discussion revolves around finding the four roots of the equation $(x-3)^4+(x-5)^4+8=0$. Participants explore the nature of the roots, particularly whether they are real or non-real, and engage in a dialogue about the implications of the equation's structure.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that there are no real roots since each term in the equation is nonnegative, leading to the conclusion that the sum cannot equal zero.
  • Others suggest that the question implies a search for non-real solutions, despite the original post not specifying this explicitly.
  • One participant provides a transformation of the equation, leading to a polynomial that suggests four complex roots, specifically $4+i$, $4-i$, $4+i\sqrt{5}$, and $4-i\sqrt{5}$.
  • Another participant humorously notes the ambiguity in the question, mentioning that the roots could theoretically include quaternions, though they agree that complex roots suffice.
  • There is a discussion about how to format solutions in replies, with participants sharing methods for using spoiler tags to hide answers.

Areas of Agreement / Disagreement

Participants generally agree that there are no real roots, but there is disagreement regarding the nature of the roots sought (real vs. non-real), and the discussion remains unresolved regarding the expectations of the original question.

Contextual Notes

The discussion highlights the ambiguity in the phrasing of the original question, which does not specify the nature of the roots, leading to varied interpretations among participants.

anemone
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Find the four roots of the equation $(x-3)^4+(x-5)^4+8=0$.
 
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anemone said:
Find the four roots of the equation $(x-3)^4+(x-5)^4+8=0$.

There aren't any real roots, each of the fourth powers is nonnegative, and so adding 8 means that it can never equal 0.

Am I correct in expecting that you wanted nonreal solutions? :P
 
Prove It said:
There aren't any real roots, each of the fourth powers is nonnegative, and so adding 8 means that it can never equal 0.

Am I correct in expecting that you wanted nonreal solutions? :P

Yes, Prove It! The question asked for the 4 non-real roots and now, I'm expecting you to solve it!(Tongueout)
 
anemone said:
Yes, Prove It! The question asked for the 4 non-real roots and now, I'm expecting you to solve it!(Tongueout)

Well, no it didn't, neither in the title nor the main body, but no matter. I'll get back to you :)
 
Prove It said:
Well, no it didn't, neither in the title nor the main body, but no matter. I'll get back to you :)

(x−3)^4+(x−5)^4+8=0.

put x - 4 = y to get

(y+1)^4 + (y-1) ^4 + 8 = 0

or 2(y^4+ 6y^2+1) + 8 = 0

or y^4 + 6y^2 + 5 = 0

(y^2 + 1)(y^2 + 5) = 0

y = i, - i, i sqrt(5), - i sqrt(5)

or x = 4+i, 4- i, 4+ i sqrt(5),4 - i sqrt(5)
 
kaliprasad said:
(x−3)^4+(x−5)^4+8=0.

put x - 4 = y to get

(y+1)^4 + (y-1) ^4 + 8 = 0

or 2(y^4+ 6y^2+1) + 8 = 0

or y^4 + 6y^2 + 5 = 0

(y^2 + 1)(y^2 + 5) = 0

y = i, - i, i sqrt(5), - i sqrt(5)

or x = 4+i, 4- i, 4+ i sqrt(5),4 - i sqrt(5)

Bravo, kaliprasad! (Clapping)And thanks for participating too...though I'd appreciate it if you would hide your solution whenever you decided to answer to any of the challenge problems...if you're okay with that, do you know how to hide your solution?
 
anemone said:
Bravo, kaliprasad! (Clapping)And thanks for participating too...though I'd appreciate it if you would hide your solution whenever you decided to answer to any of the challenge problems...if you're okay with that, do you know how to hide your solution?

I would. Could you tell me how. As a matter of fact I do not know
 
kaliprasad said:
I would. Could you tell me how. As a matter of fact I do not know

Hello kaliprasad,

The easiest way I know of to accomplish this is to compose your reply as normal, and then when you are finished, but before submitting the post, select the portion of your post that contains the actual solution using your mouse or keyboard. Then while this text is selected, click the [Sp] button on the far right of the middle row of the toolbar, and this will generate the spoiler tags to enclose the selected text. Then preview your post to make sure it looks like you intend.

Here is an image of the button to click to enclose the selected text with the spoiler tags:
View attachment 1353

Feel free to ask if you have any problems getting this to work.
 

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anemone said:
Hello kaliprasad,

The easiest way I know of to accomplish this is to compose your reply as normal, and then when you are finished, but before submitting the post, select the portion of your post that contains the actual solution using your mouse or keyboard. Then while this text is selected, click the [Sp] button on the far right of the middle row of the toolbar, and this will generate the spoiler tags to enclose the selected text. Then preview your post to make sure it looks like you intend.

Here is an image of the button to click to enclose the selected text with the spoiler tags:
View attachment 1353

Feel free to ask if you have any problems getting this to work.

Thanks

I got it if this part is enclosed else I did not
 
  • #10
Oh! And I thought the Sp button was a spell check! ;)

Seriously I didn't know that. Thanks.

-Dan
 
  • #11
Prove It said:
Well, no it didn't, neither in the title nor the main body, but no matter. I'll get back to you :)

Well, since it was not specified anywhere, the roots might also be for instance quaternions.
(I just liked saying that.)
Luckily it suffices that they are complex. ;)
To be honest, I was also confused for a moment when I realized there were no real solutions.
 
  • #12
I took the question to mean "find the 4 roots" regardless of their nature. :D
 

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