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June 6th, 2017, 12:10 PM   #1
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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?
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Last edited by skipjack; June 6th, 2017 at 02:09 PM.
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June 6th, 2017, 02:35 PM   #2
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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 left-hand 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?
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June 6th, 2017, 03:02 PM   #3
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Quote:
Originally Posted by skipjack View Post
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 left-hand 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?
Okay thanks for the reply and your question! so it was okay for me not to factor out the 3 from the 8? since I think we normally do for when it is set to zero we divide it out of all terms? and when I see a y= or a f(x) do I not see that equal sign as an = sign?

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.
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June 6th, 2017, 03:29 PM   #4
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I just don't understand why it changes from a negative on my "side" but then changes to a positive on my "other side"
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June 6th, 2017, 08:53 PM   #5
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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 re-arranging 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 re-arranging 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.
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June 7th, 2017, 05:22 AM   #6
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Quote:
Originally Posted by JeffM1 View Post
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 re-arranging 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 re-arranging 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.
THANK YOU so much so unless I got this wrong when there is a equation which means a equal sign if I add something to one side I add it to the other. IF i subtract something from one side I subtract it from the otherside. Yet with an equation if I subtract on one end, I add on the other end. But if I add on one end I subtract on the other end? I apprecate both you and skipjack. Thanks to you for not getting on me for using "sides" for both equations and expressions. You also helped me think to use "this end and this end" when it comes to expressions I will try to use that wording today and hopfully my help will understand me and my head will feel less like it wants to explode.
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June 16th, 2017, 09:55 AM   #7
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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.
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Last edited by skipjack; June 16th, 2017 at 10:08 AM.
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June 16th, 2017, 10:57 AM   #8
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I didn't see 10.

If you start with ax² + bx + c = 0, where a is non-zero,
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 left-hand side of which equals a(x + b/(2a))².

If you have the expression ax² + bx + c, where a is non-zero,
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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