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September 23rd, 2017, 01:31 AM  #1 
Newbie Joined: Sep 2017 From: UK Posts: 1 Thanks: 0  Order of Operations comprehension question
Hello, this is my first post here, so please go easy on me! Thank you for your time. As far as I understand: if $\displaystyle x = a + b$ then $\displaystyle x^{2} = (a + b)^{2}$ not $\displaystyle x^{2} = a^{2} + b^{2}$ which I can easily check by asking does $\displaystyle 5^{2} = 3^{2} + 2^{2}$? No. My question is does anyone have a way of explaining this property so that I might better understand *why* this property exists, beyond simply memorising it as a property of algebra? The reason I ask is because sometimes I make this mistake in algebraic manipulations; a specific example would be from physics: since $\displaystyle v = u + at$ $\displaystyle v^{2} = (u + at)^{2} = u^{2} + 2at + a^{2}t^{2}$ However, if I am not careful, in practise I have previously resolved this as: $\displaystyle v^{2} = u^{2} + a^{2}t^{2}$, since in my mind I am "just squaring both sides." So my request is if anyone could give me an understanding to help cement this algebraic rule in my mind in a deeper way, so I don't make this mistake again. I hope that's clear. Thanks in advance for any insight. Sam 
October 1st, 2017, 01:15 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,831 Thanks: 2161 
$(a + b)^2 = (a + b)(a + b) = a(a + b) + b(a + b) = a*a + ab + b*a + b*b = a^2 + 2ab + b^2$

October 1st, 2017, 01:20 AM  #3 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,830 Thanks: 648 Math Focus: Yet to find out. 
This mistake has a name of its own! See here. https://en.m.wikipedia.org/wiki/Freshman%27s_dream 
October 1st, 2017, 03:12 AM  #4  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Quote:
Code: 2 + 3 2 + 3  4 + 6 : start with bottom left "2" 6 + 9 : do bottom "3" next  4 +12 + 9 = 25 2 * (2 + 3), then 3 * (2 + 3)  

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