My Math Forum Squaring twice

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 January 3rd, 2017, 01:24 PM #1 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 10,682 Thanks: 700 Squaring twice I presume this is about the simplest equation where solving for x requires "squaring both sides" twice: sqrt(x) = sqrt(ux) - v A few gyrations leads to: x = [v^2 (2SQRT(u) + u + 1)] / (u^2 - 2u + 1) Is that something "known"; like, used by students as a short cut to these headachy equations? Thanks from Joppy
 January 3rd, 2017, 01:29 PM #2 Senior Member     Joined: Sep 2015 From: Southern California, USA Posts: 1,493 Thanks: 752 did you mean to reply to some thread w/this?
 January 3rd, 2017, 02:11 PM #3 Global Moderator   Joined: Dec 2006 Posts: 18,048 Thanks: 1395 If u = 1, v = 0 and x can have any value. If u isn't 1 and both x and u are non-negative, v = √(ux) - √x = (√u - 1)√x. So far, no squaring.
 January 3rd, 2017, 03:46 PM #4 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 10,682 Thanks: 700 Ya'll missing my point. I mean: when squaring both sides IS REQUIRED in order to solve. I was thinking along the lines of: students are aware that (a+b)(a-b) = a^2 - b^2; so there is no need to do the "long multiplication". I was wondering if there is a similar short cut that (at one point) students are given, in order to shorten all the work involved in squaring both sides. If I'm still not clear, forget it!
 January 3rd, 2017, 04:21 PM #5 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,352 Thanks: 463 Math Focus: Yet to find out. Won't there be too many situations when this happens though? I don't see the short cut part seems like the same amount of effort to rearrange then remember the cutting short. Thanks from Denis
 January 4th, 2017, 03:46 AM #6 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 How about you do this $\sqrt{x}( \sqrt{u} - 1) = v$ $x = ( \frac{v}{\sqrt{u} - 1} )^2$ Since $u , v$ are constants I presume , stop here then 'plug and play'
 January 4th, 2017, 03:59 AM #7 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 10,682 Thanks: 700 But that's a completely different case, Agent. Same as sqrt(x) = c (since u and v are constants/givens.) So x = c^2
 January 4th, 2017, 04:09 AM #8 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Yes but your 'gyrations' have not eliminated the square root symbol either and are more lengthy ...
 January 4th, 2017, 04:14 AM #9 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 You may be thinking of 'harder' things that make squaring twice inevitable for example ... $\sqrt{x} = \sqrt{u + x} - v$ Denis?
 January 4th, 2017, 04:30 AM #10 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 10,682 Thanks: 700 Yes, I've already stated that twice: cases where the ONLY way to solve is squaring both sides twice. Thanks from Joppy

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