Find both Real and Imaginary solutions to (x^4)-(x-1)^4=0 | Math Olympiad Preparation

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Find both Real and Imaginary solutions to (x^4)-(x-1)^4=0 | Math Olympiad Preparation

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[alittax]

Beautiful job, thank you! :) Greetings from Hungary.

[alster]

Alternate method is a combination of

Synthetic Division, Factor Theorem, and Rational Zero Theorem

[giuseppemalaguti]

Ho 3 soluzioni... 1 reale=1/2, ..2 complesse (1+i)/2 e (1-i)/2

[onlyg]

Sir I became a mathematician because of you 🥰

[RSRIAC]

It's 4 th degree polynomial where is 4th root...?

[pwmiles]

Nice problem. You can also rearrange and take the four 4th roots
x^4 = (x - 1)^4
x = x - 1, i(x - 1), - (x - 1), - i(x - 1)

x = x - 1 does not yield a solution. Otherwise solve for x
x = - i/(1 - i), 1/2, i/(1 + i)
x = - i(1 + i)/(1 - i^2), 1/2, i(1 - i)/(1 - i^2)
x = (1 - i)/2, 1/2, (1 + i)/2

[royalraj]

Here, the power on x is 4 therefore there must be 4 solutions. Then why only three solutions.

[piemotivation]

Sir I'm also doing same thing like you but my questions level is average ☺️

[aayushw]

Reincarnation of Sir Srinivasa Ramanujan

[rudychan]

In Few Secons I can Guessing: x = 1/2 !! 😉
X'''' = (X - 1)''''
X = +/- (X - 1)
X = - (X - 1) = - X + 1
X + X = 1 ---> 2X= 1 ---> X= 1/2
The Only Solution is 0, 5 !! 🤓
√-1 is Not Exist !! An Error !!

Well, Anyway, I Give You Thumb Up 👍 31st

[alcibiadesmarcialneto]

A 4th degree equation demands 4 solutions. You miss x=-1/2.