If 2x+2^{2y}=2x-2z^2 what is z in terms of y?

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Answer 1 · 2017-04-21T09:00:44

#z=+-sqrt(-2^((2y-1)))#, but there is no rational solution.

As # 2x+2^(2y)=2x-2z^2#, we have

#2^(2y)=-2z^2#

or #2×2^((2y-1))=2×(-z^2)#

or #2^((2y-1))=-z^2#

or #z^2=-2^((2y-1))#

or #z=+-sqrt(-2^((2y-1)))#

As a power of #2# is always positive, #z# has no rational solution.

Answer 2 · 2017-04-21T09:05:24

# pmsqrt(2)/2 2^y i#

You have

#2x+2^{2y}=2x-2z^2# so adding #-2x# to both equation sides

#-2x+2x+2^{2y}=-2x+2x-2z^2# so

#2^(2y)=-2z^2# or

#z^2= -1/2 2^(2y) = -2^(2y-1)# and finally after extracting the square root to both sides

#z = pm sqrt(-2^(2y-1))= pm i 2^y2^(-1/2)= pmsqrt(2)/2 2^y i#

here #i = sqrt(-1)#