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x⁴ = (x - 2)⁴
Full binomial expansion and factoring:
On both sides, drop x⁴ for cubic; 3 solutions.
x⁴ = x⁴ - 4*x³*2 + 6*x²*4 - 4*x*8 + 16
0 = -8*(x³ - 3*x² + 4*x - 2)
x³ - 3*x² + 4*x - 2 = 0
(x - 1)*(x² -2*x + 2) = 0
x = 1, 1 ± i
Factoring difference of squares:
x⁴ - (x - 2)⁴ = 0
[x² - (x - 2)²]*[x² + (x - 2)²] = 0
[x - (x - 2)]*[x + (x - 2)]*[2*x² - 4*x + 4] = 0
[2]*[2*x - 2]*[2]*[x² - 2*x + 2] = 0
8*[x - 1]*[(x - 1)² + 1] = 0
x = 1, 1 ± i
Roots of 1:
((x - 2)/x)⁴ = 1 = r⁴ , where r = ±1, ±i
1 - 2/x = r
x = 2/(1 - r)
= 2/0, 2/2, 2/(1 - i), 2/(1 + i)
= ∅, 1, 1 + i, 1 - i
x = 1, 1 ± i