What are the four roots of this challenging equation?

In summary: Luckily it suffices that they are complex. ;)Thank you, that clears things up a bit. :)In summary, there are no real roots of the equation $(x-3)^4+(x-5)^4+8=0$.
  • #1
anemone
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Find the four roots of the equation $(x-3)^4+(x-5)^4+8=0$.
 
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  • #2
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
 
  • #3
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)
 
  • #4
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 :)
 
  • #5
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)
 
  • #6
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?
 
  • #7
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
 
  • #8
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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  • #9
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
 

FAQ: What are the four roots of this challenging equation?

What is the "Roots Finding Challenge"?

The "Roots Finding Challenge" is a mathematical problem that involves finding the values of a variable that make a given equation equal to zero. The solutions to this problem are known as roots.

Why is it important to find roots in mathematics?

Finding roots is important in various fields of mathematics, such as algebra, calculus, and statistics. It allows us to solve equations, understand the behavior of functions, and make predictions based on data analysis.

What methods are commonly used to find roots?

There are several methods used to find roots, including the bisection method, Newton's method, and the secant method. These methods involve making successive estimations of the roots until a desired level of accuracy is achieved.

What are some challenges in finding roots?

One of the main challenges in finding roots is dealing with complex equations that do not have analytical solutions. In these cases, numerical methods must be used, which can be time-consuming and require a good understanding of the problem at hand.

How can the Roots Finding Challenge be applied in real life?

The Roots Finding Challenge has many practical applications, such as in engineering, physics, and finance. For example, it can be used to calculate the trajectory of a projectile, determine optimal investment strategies, or model the spread of diseases.

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