SOLUTION: the maximum value of |z| when z satisfies the condition |z+2/z|=2 is

Question 887983: the maximum value of |z| when z satisfies the condition |z+2/z|=2 is
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
we have two cases to consider
1) z +2/z = 2
multiply both sides of = by z (note that z not = 0)
z^2 +2 = 2z
z^2 -2z = -2
complete the square
z^2 -2z +1 = -2 +1
(z -1)^2 = -1
z -1 = square root(-1)
z = 1 +i (note that square root -1 is i)
2) -(z +2/z) = 2
multiply both sides of = by -
z +2/z = -2
multiply both sides of = by z
z^2 +2 = -2z
z^2 +2z = -2
complete the square
z^2 +2z +1 = -2 +1
(z +1)^2 = -1
take square root of both sides of =
z +1 = square root(-1)
z = i - 1
therefore z = i-1 or 1+i
note that |a +bi | = square root(a^2 +b^2)
|z| = |i - 1| = square root(1 +1) = square root(2)
|z| = (1 +i| = square root(1 +1) = square root(2)
|z| = square root(2) or -square root(2)
we choose the positive square root(2)
|z| = square root(2)

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