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Factoring: Grouping, Perfect Squares, Finding Domains, Solving Equations |Fast Track Math
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Feb 20, 2025
In this 9 minutes videos , my main emphasis is factoring. i was helping one of my students and we worked on factoring using the grouping methos, using identities. We also used factoring to solve rational equations and finally we also applied it to find the domain of a rational function. I also introduced the Fast Track Math book that I used a couple of months ago to help some of my college students. A lot of methods that I used in the video were taken from the book which I highly recommend. If you're interested in it shoot me an email via [email protected]
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0:00
basically we are trying to factor 6x
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squared plus 17x last hang right so we
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take the six and ten and we multiply
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them together six times ten I give you
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six sixteen right so 6 times 10 I give
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you 60 the next step is you want two
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numbers whose product is 60 and Sony 17
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right so now we can victor on 60 60 is 6
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and 10 60 is also 5 and 12 right now 6
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plus 10 gives me 16 so I'm not going to
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deal with 12 17 yes so now I'm going to
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rewrite this whole operation so this is
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6x squared plus 17x right Stan right I'm
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gonna rewrite it we're using my digit
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yeah right so it's gonna be 6x squared
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plus 12x right plus 5x plus 2 again I'm
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not I'm not touching this yet
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I just split it so I can I can do my
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factorization using the grouping right
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okay so I'm gonna take these two
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together
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and I'm gonna take these two together
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right what's the common factor between
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6x squared X 6 X right if a lot is 6x
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out of this I'm left with 2x right now
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I'm gonna sweep X I'm sorry X right
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that's two right and it's uploaded 5 out
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of here
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I'm gonna move X plus 2 now you see we
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have the common factor the same thing X
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plus 2 and X plus 2 so therefore this
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becomes X plus 2 times 6x plus 5 and
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you're done all right
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so use the grouping I don't like the
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guessing check because the guessing
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check is a lot of work you just keep
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trying you want to have a practical way
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doing it so you split the two and then
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find a common factor by grouping
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okay okay our next step is we trying to
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factor 6x squared minus 54 now this is
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this has like two expression in there
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two times right 6x squared and 54 now
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this is probably we do not use it what
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we call the difference of two squares
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right now both numbers have to be
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perfect squares perfect square is like 9
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is a perfect square because 3 times 3 is
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9 16 is a perfect square
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well when I look at this expression I
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don't have a perfect square but I do
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have 6 and I do have to define unnoticed
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6 goes into 6 and 6 goes into 54 so I'm
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gonna pull out is 6 as a factor right
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I'm gonna be left with x squared minus
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1/9 because 6 times 9 is right and then
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now I have my perfect square because I
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have x squared minus 9 right because 9
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is a perfect square and x squared -
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perfect spray and I know that a square
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right minus B squared is a minus B right
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and then a plus B that's an identity
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right so it's gonna be 6 we keep a 62
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expression alright so the key here was
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to recognize that 6 goes into 6 and 54
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at the same time in factor out the
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common factor and then you get your
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identity and then use the difference of
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two perfect squares alright alright so
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we're trying to find a domain or this
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function right it's a rational function
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so we have to look at the denominator
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any rational function cannot have a
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denominator equal to zero because you
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know a number of a zero is undefined so
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we don't want zero to be in the
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denominator so therefore the domain is
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we don't want this expression to be
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equal to zero right
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and we're gonna solve for X so then we
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do in factoring so therefore I'm gonna
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put our X right X plus 8 not equal to
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zero
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all right I'm going to go ahead and
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finish it so it's going to be X not
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equal to zero
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you split them right
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Plus eight not equal to zero so
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therefore X not equal to negative eight
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right so my denominator will be every
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number except negative eight and what
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zero which is gonna be my domain I don't
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want no it's not negatively and zeros
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all real numbers except zero and
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negative is how you write it all real
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numbers right except zero and negative
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eight or you could just go R minus zero
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and eight is how you can write it if you
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want but if you you you you just say all
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real numbers except zero negative it
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everything was even negative eight you
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know what everybody did not add the
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numerator just a denominator okay for
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the domain of a rational function okay
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alright so the next thing is we're
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trying to solve this equation so now we
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are in doing factoring right this is
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what we're doing so therefore I'm going
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to look at this X minus 2x x squared
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minus 4x plus 2 right now I'll recognize
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something here this is the difference of
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two squares so I'm gonna rewrite this
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operation as 1 over X minus 2 minus 3 X
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and then X plus 2 X minus 2 equals 2 2 X
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plus 2 right now I know one thing I want
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to find my lowest common denominator
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right and the lowest common denominator
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is gonna be this whole thing here so
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then to make this easier we're gonna
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multiply every single entry by this
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[Music]
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simplified right so it's going to be X
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plus 2 X minus 2 1 over X minus 2 minus
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X plus 2 X minus 2 times 3x right over X
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plus 2 exponent from here okay I keep
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you doing you're at the top all right
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and then we do the same thing here and
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you see what's gonna happen it's gonna
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simplify the whole cancellous right
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these
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goes away and then this goes away right
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you left with X plus 2 right here times
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1 so we just 1 is the same thing right
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anytime Center 1 minus 3 X right equals
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2 2 right I'm X minus 2 okay and IRS is
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just a piece of cake you can go ahead
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and just come by like that I'll let you
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finish that hello this is ty even
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actually you saw me working on some
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factoring problem earlier and a lot of
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the things that I teach in mathematics
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I've actually thought by using this book
7:10
it's called the basic Fastback and this
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is the second edition now two months ago
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I work with like 9 students for like a
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week and a half and we actually went and
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covered the entire pre-algebra /
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elementary algebra topics we did
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rounding numbers multiplication
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notations factoring a whole lot of stuff
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it was crammed into a week and a half
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actually two weeks to be more precise
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and I've used this book I mean it's very
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concise and he goes through it to the
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point you're not gonna wish the time
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you've lengthy stuff that you don't need
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to know and I'm a math tutor / I got my
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degree in math as well so I know what
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I'm talking about I'm not just gonna
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give you something that's not practical
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you know a lot of time you hear people
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talk about doing things a certain way
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and that's like ways of solving a
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problem and for me it's like this which
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ever worked for the students is what I
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want to stick to and this book here
8:04
gives you a lot of options and I've
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actually used it we must begin like I
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said and he was really good like they
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came out of lectures really happy and
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ready to like taku any topics so that
8:20
means so concise and and let me show you
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the back of the book and you can kind of
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read this yourself if you want right so
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the goal is to save you time and money
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by placing you in the highest level
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course there's a lot of time you go to
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like a college when they make you take
8:37
all this like elementary math classes
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and you're wasting money and you're
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wasting time so with this book it gives
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you almost like
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you when when you when you take the
8:46
Accuplacer for some colleges or whatever
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might have to has two placement tests
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you know these things are already so
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you're not gonna test out of you're not
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gonna test
8:54
you're not gonna pitch in like low level
8:56
courses especially if you've been out of
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college for a while and you want to go
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back and then you don't want to waste
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your time on basic stuff so this book is
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definitely I highly recommend it I'm
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just letting you know I've used it with
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my students I've seen the progress and I
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like it I mean there's many books out
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there that I've seen in the past where
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this one was the best out of all of them
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all right so if you want to just shoot
9:19
me an email and I'll show you how to get
9:21
it thanks