Sketch a GraphDate: 6/1/96 at 21:56:38 From: Anonymous Subject: Sketch a graph z=(3-2i)^1/2 then find z^- . Sketch an appropriate graph to illistrate z^- and z if z is in quadrant 1. Show and describe how they are related geometrically. Date: 6/2/96 at 7:37:40 From: Doctor Anthony Subject: Re: Sketch a graph Let z = a+ib then z^2 = a^2-b^2 + 2iab = 3-2i Equate real and imaginary parts and we have two equations for finding a and b. a^2-b^2 = 3 2ab = -2 so ab = -1 and b = -1/a Substitute for b in the first equation a^2 - 1/a^2 = 3 multiply through by a^2 a^4 - 1 = 3a^2 so a^4 - 3a^2 - 1 = 0 This is a quadratic in a^2, and using the quadratic formula we have a^2 = {3 +or-sqrt(9+4)}/2 We must take the positive square root since a^2 must be positive. a^2 = {3+sqrt(13)}/2 = 5.408 and a = 2.32558 b = - 0.43 So z = 2.3256 - 0.43i conjugate z will be 2.3256 + 0.43i z^- is simply the reflection of z in the x axis. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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