What are the four roots of this challenging equation?

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SUMMARY

The equation $(x-3)^4+(x-5)^4+8=0$ has no real roots due to the nonnegative nature of the fourth powers and the addition of 8, which ensures the expression cannot equal zero. The four complex roots are derived by substituting \(x - 4 = y\), leading to the equation \(y^4 + 6y^2 + 5 = 0\). The solutions are \(y = i, -i, i\sqrt{5}, -i\sqrt{5}\), which correspond to the roots \(x = 4+i, 4-i, 4+i\sqrt{5}, 4-i\sqrt{5}\).

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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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