Roots Finding Challenge

anemone

MHB POTW Director
Staff member
Find the four roots of the equation $(x-3)^4+(x-5)^4+8=0$.

Prove It

Well-known member
MHB Math Helper
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?

anemone

MHB POTW Director
Staff member
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?
Yes, Prove It! The question asked for the 4 non-real roots and now, I'm expecting you to solve it!

Prove It

Well-known member
MHB Math Helper
Yes, Prove It! The question asked for the 4 non-real roots and now, I'm expecting you to solve it!
Well, no it didn't, neither in the title nor the main body, but no matter. I'll get back to you

Well-known member
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)

anemone

MHB POTW Director
Staff member
(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! 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?

Well-known member
Bravo, kaliprasad! 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

anemone

MHB POTW Director
Staff member
I would. Could you tell me how. As a matter of fact I do not know

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
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:

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

Well-known member

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
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

topsquark

Well-known member
MHB Math Helper
Oh! And I thought the Sp button was a spell check!

Seriously I didn't know that. Thanks.

-Dan

Klaas van Aarsen

MHB Seeker
Staff member
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.