SOLUTION: Solve (2x-1)^4 + (2x-1)^2 - 6 = 0

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Question 1116475: Solve (2x-1)^4 + (2x-1)^2 - 6 = 0
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
+%282x-1%29%5E4+%2B+%282x-1%29%5E2+-+6+=+0+
Let +u+=+%282x-1%29%5E2+
+u%5E2+%2B+u+-+6+=+0+
+%28u-2%29%28u%2B3%29+=+0+
u=2 and u=-3 are solutions, consider each in turn:

u=2: +%282x-1%29%5E2+=+2+
+4x%5E2+-+4x+%2B+1+=+2+
+4x%5E2+-+4x+-+1++=+0+
+x%5E2+-+x+-+%281%2F4%29+=+0+
+x%5E2+-+x+++=+%281%2F4%29+
+x%5E2+-+x+%2B+%281%2F4%29++=+%281%2F4%29+%2B+%281%2F4%29+
+++%28x+-+1%2F2%29%5E2++=+1%2F2+
++x+-+1%2F2++ = +/- +1%2Fsqrt%282%29++
++x+ = +/- +sqrt%282%29%2F2+%2B+%281%2F2%29+

Two of the four solutions are real solutions:
+highlight%28++x=+%281-sqrt%282%29%29%2F2+%29+ and highlight%28+x=%281%2Bsqrt%282%29%29%2F2+%29+

Now for the complex solutions:
u=-3: +%282x-1%29%5E2+=+-3+
++4x%5E2+-+4x+%2B+1+=+-3+
++4x%5E2+-+4x++=+-4+
++x%5E2+-+x+=+-1+
+x%5E2+-x+%2B+1%2F4+=+-3%2F4+
+%28x-1%2F2%29%5E2+=+-3%2F4+
+++x-1%2F2+ = +/- +sqrt%283%29i%2F2+

Which has complex solutions:
++highlight%28+x=+1%2F2+-+i%2A%28sqrt%283%29%2F2%29+%29+ and +highlight%28+x=1%2F2+%2Bi%2A%28sqrt%283%29%2F2%29%29+

The four roots together are all of the solutions to the 4th order equation.