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June 6th, 2017, 12:10 PM  #1 
Senior Member Joined: Feb 2016 From: seattle Posts: 377 Thanks: 10  trying to get the wording for completing the square (please delete if not allowed)
I tried to find somewhere else on this site or online to place this but no luck if I can possibly get it answered here or someone willing to message me about it or where to find such a place. My lack of knowing what to say is really upsetting me and confusing me as when I say something my lack of "correct words" is limiting my learning and my helps understanding me. They just don't seem to get what I am saying. It seems to me that when I have a question asking like in the first question our squared number gets added to both sides of our equal sign. Then in the second question because of the 3 I feel we do something different other than just making it negative, (but before I add to that if I am able to should this post still be around) should I have also gotten 8/3 for the 8 or it's okay to not divide the 3 out of it? Last edited by skipjack; June 6th, 2017 at 02:09 PM. 
June 6th, 2017, 02:35 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,843 Thanks: 1565 
Whatever is added to one side of the "=" must also be added to the other side. For question 1, 9/4 should be added to both sides, because the lefthand side will then be x² + 3x + 9/4, which is (x + 3/2)². For question 2, there is no "=", but "3(x² + 16x + 64) + 8 + 192" would be an equivalent expression and then "3(x² + 16x + 64)" can be replaced with the equivalent "3(x + 8)²". Take care to get the signs correct. Do you need further information on this type of problem? 
June 6th, 2017, 03:02 PM  #3  
Senior Member Joined: Feb 2016 From: seattle Posts: 377 Thanks: 10  Quote:
also I am getting confused since when we multiply the 3(64) it gets us 192 so how come (when we add) it (move It over ) or what hopefully is less incorrect wording it became positive? This is where my help and I both loose being able to communicate when I am doing y= or a f(x) or one like question 2 she does not see it as "sides since it's all over on one side" but to me I always want to say the other side as in we have the 192 on this side of the ")" or the "8" then we have this side over here so to me that is two sides as well , not in the someway as on = sign sides but.... Last edited by greg1313; June 6th, 2017 at 06:45 PM.  
June 6th, 2017, 03:29 PM  #4 
Senior Member Joined: Feb 2016 From: seattle Posts: 377 Thanks: 10 
I just don't understand why it changes from a negative on my "side" but then changes to a positive on my "other side"

June 6th, 2017, 08:53 PM  #5 
Senior Member Joined: May 2016 From: USA Posts: 993 Thanks: 407 
Well, let's deal with vocabulary and purpose. Your first problem deals with an equation because it has an equal sign in it. We frequently try to solve equations, meaning to find one or more numbers that make the equation true. Your second problem as you have stated it deals with an expression, not an equation, because there is no equal sign in it. All you can do with an expression is to find another expression that is equivalent to the first for all numbers. The purpose of completing the square is to turn a quadratic expression into an equivalent expression that contains a perfect square. So let's take your second problem. $\ 3x^2  48x + 8$ is an expression. We want to find an equivalent expression that contains a perfect square. $\ 3x^2  48x + 8 \equiv (\ 3)(x^2 + 16x) + 8.$ Do you agree that the equivalence above is true? Whatever number x is, both sides of the equivalence give the same result, right? We are NOT changing the value of the expression, just changing its form. That is why I used $\equiv.$ $(\ 3)(x^2 + 16x) + 8 \equiv (\ 3)(x^2 + 2 * \dfrac{16}{2} * x) + 8.$ Again we have changed form but not the value, correct? $(\ 3)(x^2 + 2 * \dfrac{16}{2} * x) + 8 \equiv (\ 3)(x^2 + 2 * 8 * x) + 8.$ No change in value, right? $(\ 3)(x^2 + 2 * 8 * x) + 8 \equiv (\ 3)(x^2 + 2 * 8 * x + 8^2  8^2) + 8.$ Different form, same value. $(\ 3)(x^2 + 2 * 8 * x + 8^2  8^2) + 8 \equiv (\ 3)(x^2 + 2 * 8 * x + 8^2  64) + 8.$ No change in value whatever x is. $(\ 3)(x^2 + 2 * 8 * x + 8^2  64) + 8 \equiv (\ 3)(x^2 + 2 * 8 * x + 8^2) + (\ 3)(\ 64) + 8.$ Have we changed the value? No. $(\ 3)(x^2 + 2 * 8 * x + 8^2) + (\ 3)(\ 64) + 8 \equiv (\ 3)(x^2 + 2 * 8 * x + 8^2) + 192 + 8.$ Still no change in value. $(\ 3)(x^2 + 2 * 8 * x + 8^2) + 192 + 8 \equiv (\ 3)(x^2 + 2 * 8 * x + 8^2) + 200.$ No change in value. $(\ 3)(x^2 + 2 * 8 * x + 8^2) + 200 \equiv (\ 3)(x + 8 )^2 + 200.$ I have gone through this in baby steps to make clear that when rearranging an expression, you cannot do anything that changes its numeric value. Clear on that? However, with an equation, which has two expressions joined by an equal sign, you can and should do things that change the value of an expression on one side provided that you change the value of the expression on the other side of the equation the exact same way. When rearranging an expression, you must maintain the original value. When solving an equation, you will change the original values but must maintain the equality by changing both expressions the exact same way. Usually on the first day or so in algebra they briefly mention the difference between an expression and an equation, but don't really explain that you work with them in different ways. 
June 7th, 2017, 05:22 AM  #6  
Senior Member Joined: Feb 2016 From: seattle Posts: 377 Thanks: 10  Quote:
 
June 16th, 2017, 09:55 AM  #7 
Senior Member Joined: Feb 2016 From: seattle Posts: 377 Thanks: 10 
Sadly, this is still keeping me back; I feel like I am getting close, but I don't understand how it got bigger? How did 5 become 10? and where did the 1 half come from? Thanks. Last edited by skipjack; June 16th, 2017 at 10:08 AM. 
June 16th, 2017, 10:57 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 18,843 Thanks: 1565 
I didn't see 10. If you start with ax² + bx + c = 0, where a is nonzero, it can be written as a(x² + (b/a)x) + c = 0, and adding b²/(4a)  c to both sides gives a(x² + (b/a)x + b²/(4a²)) = b²/(4a)  c, the lefthand side of which equals a(x + b/(2a))². If you have the expression ax² + bx + c, where a is nonzero, write it as a(x² + (b/a)x + (b/(2a))²)  a(b/(2a))² + c, which equals a(x + (b/(2a))²  b²/(4a) + c. 

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