Solve the equation `(x-1)^4+(x-5)^4=82.`

Explanation

Let `y = ((x - 1) + (x - 5)) /(2) = x - 3`
`rArr x = y + 3`
Putting x = y + 3 in the given equation, we obtain
` (y+ 2)^(4) + (y - 2)^(4) = 82`
or `(y^(2) + 4y + 4)^(2) + (y^(2) - 4y + 4)^(2) = 82`
or `{(y^(2) + 4)^(2) + 4y}^(2) + {(y^(2) + 4) + 4y}^(2) = 82`
or `2{(y^(2) + 4)^(2)+ 16y ^(2)} = 82`
`[therefore(a + b )^(2)+ (a - b)^(2) = 2(a^(2)+ b^(2))]`
or `Y^(4) + 8y^(2) + 16 + 16y^(2) = 41`
or `y^(4) + 24y^(2) - 25 = 0`
or `(y^(2) + 25) (y^(2) - 1) = 0`
or `y^(2) + 25 = 0, y^(2) - 1 = 0`
`rArr y = pm 5i, y = pm1` ("where" `i = sqrt(-1))`
`rArr x - 3 = pm 5i, x - 3 = pm 1`
or `x = 3 pm 5i, x = 4, 2`
Hencem the roots of the given equation are `3pm 5i, 2 ` and 4.