SOLUTION: solve: (x+iy)^5 + (x-iy)^5 = -8

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Question 1143530: solve: (x+iy)^5 + (x-iy)^5 = -8
Found 3 solutions by greenestamps, Alan3354, ikleyn:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!

There is one equation in two variables; it can't be SOLVED. You can only expand the binomials and SIMPLIFY the expression on the left.

Use the 5th row of Pascal's Triangle and the binomial theorem to expand both terms.
(x+iy)^5 = x^5 + 5x^4y(i) + 10 x^3y^2(i^2) + 10 x^2y^3(i^3) + 5xy^4(i^4) + y^5(i^5) + (x-iy)^5 = x^5 + 5x^4(-y)(i) + 10 x^3(-y)^2(i^2) + 10 x^2(-y)^3(i^3) + 5x(-y)^4(i^4) + (-y)^5(i^5) -------------------------------------------------------------------------------------------------------------- 2x^5 + 20x^3y^2 + 10xy^4

It makes no sense to say to set that expression equal to -8 and ask to solve the equation....

Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website!
solve: (x+iy)^5 + (x-iy)^5 = -8
-----------
C = x + iy ---> sqrt(2)cis(45)
D = x - iy ---> sqrt(2)cis(315)
===========================
C^5 = sqrt(32)cis(225)
D^5 = sqrt(32)cis(1575) = sqrt(32)cis(135)
-8 = -8cis(0) = -8*1 + i*0
-----------------
C^5 = sqrt(32)*(-sqrt(2)/2 + i*(-sqrt(2)/2))
D^5 = sqrt(32)*(-sqrt(2)/2 + i*(sqrt(2)/2))
-----
C^5 + D^5 = -sqrt(64) = -8
----------------------------
Did you mean solve?
Or confirm the identity?
===================================
It's a confirmation of the case that x & y are equal. I forgot to state that.
If x <> y, then it's more tedious and not worth the effort without knowing what you want to do.

Answer by ikleyn(52780)   (Show Source):
You can put this solution on YOUR website!
.

The post by Alan is NOT a solution, because it contains mistakes.

First mistake is in its first line " C = x + iy ---> sqrt(2)cis(45) ". It is an assumption, but not an identity. Second mistake is in its second line " D = x - iy ---> sqrt(2)cis(315) ". Again, it is an assumption, but not an identity.

Actually, "the solution" to this equation is some CURVE line at the coordinate plane (x,y).