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Sir/Madam, can I please get the answer for the following questions:
1.) (x-1)^4+(x-5)^4=82
Solve for x.
2.) (5+2√6)^(x^2-3)+(5-2√6)^(x^2-3)=10
Solve for x.
10 months ago

Dear student

Please only one question in one thread.

1.)

It's easy to check that x = 4, 2 are two solutions to (x - 1)^4 + (x - 5)^4 = 82.
Now, we can easily find the other values of x.
(x - 1)^4 + (x - 5)^4 = 82
==> (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
10 months ago
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