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March 12th, 2011, 07:48 AM  #1 
Senior Member Joined: Dec 2010 Posts: 233 Thanks: 0  Number and sets
Given (a+bi)^2=2i. Find all the real values of a and b. The solution (a+bi)^2=2i a^2b^2+2abi=2i Equating real parts: a^2b^2=0 a^2=b^2 a=+ b Equating the imaginary part: 2ab=2 ab=1 Please explain to me about this part below. I am confused. Thank you. When a=b, a^2=1 (I still don't understand how to get this) a=+1 b=+1 When a=b, a^2=1 a^2=1 (rejected) because a should be a real number The real values of a and b are 1 and 1, 1 and 1. 
March 12th, 2011, 11:56 AM  #2 
Senior Member Joined: Nov 2010 From: Staten Island, NY Posts: 152 Thanks: 0  Re: Number and sets
From 2ab=2 substitute a in for b (because a=b). So we get 2a*a=2.

March 12th, 2011, 12:06 PM  #3 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Number and sets
You found that ab = 1. Then when a = b, we rewrite the above as a(a) = 1 after substituting a in place of b. a^2 = 1 ... 
March 12th, 2011, 11:38 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,114 Thanks: 1909 
Using (1 + i)² = 1 + 2i  1 = 2i, the original equation becomes (a + bi)² = (1 + i)², so a + bi = ±(1 + i). If a and b are real, equating real and imaginary parts gives a = b = ±1. 
March 13th, 2011, 12:00 AM  #5 
Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0 
(a + bi)² = 2i a² + 2abi  b² = 2i a²  b² = 0, ab = 1 a²  1/a² = 0 a² = 1/a² a^4 = 1 ? a = ±1 b = 1/a = 1/±1 = ±1 (? 1/1 = 1, 1/1 = 1) ? a = b = ±1 

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