My Math Forum Question about complex absolute value

 Pre-Calculus Pre-Calculus Math Forum

 April 27th, 2019, 10:56 AM #1 Senior Member     Joined: Oct 2016 From: Arizona Posts: 209 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems. Question about complex absolute value If $z$ is a complex number and $|z^2-1|=|4z-i|$ can I go ahead and say $$z^2-1=4z-i$$? I'm using this to prove $|z|^2 \le 9+\sqrt{85}$. I'm not sure what steps to take other than what I mentioned. PLEASE just give a hint. Do not post a full solution, just please tell me how to proceed. Last edited by ProofOfALifetime; April 27th, 2019 at 11:07 AM.
 April 27th, 2019, 11:34 AM #2 Senior Member     Joined: Oct 2016 From: Arizona Posts: 209 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems. Oh.. I was able to get it. Nevermind! Tricky but cool problem! Thanks from idontknow
April 27th, 2019, 12:36 PM   #3
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Quote:
 Originally Posted by ProofOfALifetime If $z$ is a complex number and $|z^2-1|=|4z-i|$ can I go ahead and say $$z^2-1=4z-i$$?
Not always .

 April 27th, 2019, 01:39 PM #4 Senior Member     Joined: Oct 2016 From: Arizona Posts: 209 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems. I wanted to avoid using $z=a+bi$ but I don't think it's possible.
 April 27th, 2019, 06:50 PM #5 Senior Member     Joined: Oct 2016 From: Arizona Posts: 209 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems. This problem is much harder than I thought!
 April 28th, 2019, 07:35 AM #6 Senior Member     Joined: Oct 2016 From: Arizona Posts: 209 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems. Actually, it was hard because I messed up on a part of it so that problem no longer made sense. I finally solved it!
 April 29th, 2019, 02:12 PM #7 Senior Member     Joined: Oct 2016 From: Arizona Posts: 209 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems. OMG if anyone wants to do this problem, I encourage it! I finally solved it and my goodness it was like a journey! I don't know about you all, but I found this problem challenging but the solution rewarding! SORRY for spamming the thread! NOTE:If you do solve it, DO NOT post a solution since it is from a math journal and technically isn't supposed to be posted online, but DEFINITELY do not post a the solution! Thanks! You guys rock! Last edited by ProofOfALifetime; April 29th, 2019 at 02:26 PM.

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