Other approach: Multiply both sides by 2⁶=64, substitute t=2x-1 and get (t+1)⁶=(t-1)⁶. After expanding, reducing and factoring we get the quintic in t: t(3t²+1)(t²+3)=0 with roots t=0, t=±(√3/3)i, t=±√3i. Finally x=(t+1)/2 and x=1/2, x=(3±√3i)/6, x=(1±√3i)/2
9 часов назад+1
great. The unique real solution 1/2 is obvious by the symmetry of the problem, so the substitution makes everything flow smoothly
In the first method, where you used the quadratic formula for the 3rd factor, the denominator should have been 2a = 6.
The solution to this equation may be written in a compact form as:
x = 1/2 + i (n/6) √3 , {n} = {0 , ± 1 , ± 3}
Other approach:
Multiply both sides by 2⁶=64, substitute t=2x-1 and get (t+1)⁶=(t-1)⁶.
After expanding, reducing and factoring we get the quintic in t: t(3t²+1)(t²+3)=0 with roots t=0, t=±(√3/3)i, t=±√3i.
Finally x=(t+1)/2 and x=1/2, x=(3±√3i)/6, x=(1±√3i)/2
great. The unique real solution 1/2 is obvious by the symmetry of the problem, so the substitution makes everything flow smoothly
problem
x⁶ = (x-1)⁶
(x-1)⁶ - x⁶ = 0
This is quintic because the x⁶ subtracts off of the expansion of (x-1)⁶, so we expect 5 roots.
Factor as a difference of 2 cubes.
[ (x-1)² - x²] [ (x-1)⁴ +(x-1)² x² +x⁴] = 0
Use the zero product property.
First term:
(x-1)² - x² = 0
(x-1-x)(x-1+x) = 0
2x = 1
x = 1/2
Second term:
(x-1)⁴ +(x-1)² x² +x⁴ = 0
(x-1)² [ (x-1)² + x²] + x⁴ = 0
(x²-2x+1)(2x²-2x+1) + x⁴ = 0
3(x²-x)² + 4(x²-x) + 1 = 0
Let
y = x²-x
3y² +4y + 1 = 0
y = [ -4 ± √4 ] / 6
= ( -2 ± 1 ) / 3
= -1, -1/3
x²-x = -1
(x - 1/2) ²-1/4 = -1
x -1/2 = ± i√3 /2
x = 1/2 ± i√3 /2
x²-x = -1/3
(x - 1/2) ²-1/4 = -1/3
(x - 1/2) ² = -1/12
x - 1/2 = ± i √3 / 6
x = 1/2 ± i √3 / 6
answer
x ∈ { 1/2,
(1 - i√3) / 2,
(3 - i√3) / 6,
(3 + i√3) / 6,
(1 + i√3) / 2 }