1. Solve for real values of x :- (x-1)4 + (x-5)4 = 82

Asked by Prince Sonu | 18th Apr, 2013, 01:22: PM

Expert Answer:

(x - 1)^4 + (x - 5)^4 = 82
 
Hence, here also applying the same formula
==> (x^4 - 4x^3 + 6x^2 - 4x + 1) + (x^4 - 20x^3 + 150x^2 - 500x + 625) = 82
==> 2x^4 - 24x^3 + 156x^2 - 504x + 544 = 0
==> x^4 - 12x^3 + 78x^2 - 252x + 272 = 0.

By long/synthetic division,
x^4 - 12x^3 + 78x^2 - 252x + 272 
= (x - 4) (x^3 - 8x^2 + 46x - 68)
= (x - 4)(x - 2)(x^2 - 6x + 34).

So, we have
(x - 4)(x - 2)(x^2 - 6x + 34) = 0
==> x = 4, 2, or x = 3 ± 5i
 
So, the real solutions are x=4 and x=2

Answered by  | 18th Apr, 2013, 09:10: PM

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