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How To Write & Solve Quadratic Equations By Factoring
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Feb 19, 2025
In Chapter 4.3, we'll learn how to write quadratic equations in the standard from and also solve them by way of factoring. Chapters: 00:00 Introduction 00:37 Standard form of any quadratic equation 01:02 How to write a quadratic equation given its roots 08:48 How To solve a quadratic equation by factoring the GCF 12:56 How to solve a quadratic equation using the perfect square 21:18 How to solve a quadratic equation using the difference of two squares
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0:01
so this is uh I skipped on purpose section 4.2 because is irrelevant
0:07
section 4.2 deals with like finding solving quadratic equations by graphing
0:12
it's just not that great so I was like let's skip that let go straight to 4.3
0:18
where we actually going to learn how to solve equations by factoring right now
0:24
remember I first mentioned the standard uh form of any quadratic equation the
0:30
standard form is always a X2 + b x + C that's the standard right now suppose
0:39
that I give you a quadratic equation and I want you to find the equation in its standard form
0:45
so how do you do that right so this is the way you do it first we need to find
0:51
your zeros right the zeros of a quadratic equation is where the function
0:57
crosses what axis X or Y the what the zeros x- axis right anytime
1:04
you have the function crossing the x- axis you have what you called a zero right zero meaning at that specific
1:11
point the value of a function is actually what zero right so given a zero
1:16
of a function you can build the function if I don't have the function per se
1:22
right if I have this but I do not have the actual function how can I find the
1:28
equation of the function given the the graph the zeros come in handy
1:34
right because if you have an equation if you have a function and you want to find its equation all you have to do is just
1:40
use this especially this is quadratic because any quadratic equation is always in the second degree which mean the
1:46
highest exponent is x² so if I give you a function and I want you to build this
1:53
equation you're going to use this formula here really simple I know you guys love formulas so if you formula
2:00
person this is going to be handy right so example 1.1 say translate sentences into
2:08
equation so I want you to write a quadratic equation in the standard form
2:13
with roots 1/3 and six the word root right r w TS and zeros
2:22
are synonymous it means the same thing same thing right so if I'm giving the
2:27
root or the Z how do I find a function I'm going to use this formula here right
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so in our case we can call this one X1 and we can call this one X2 the only
2:40
thing that I don't have is a the value of a now if I don't specify anything to
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you you can use any values for a you can use 1 2 3 4 5 6 7 8 whatever number you
2:50
want to use right cuz I'm asking you to find write a quadratic equation I didn't say right the I didn't specify a
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specific when one I say a quadratic equation that means the value of a is irrelevant you can choose whatever value
3:04
you want for a does that make sense right so for for this case we're going to use a = to 1 right I'm going to make
3:11
it easy so now I'm going to
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yeah oh yeah sure so one you want to PR second the in
3:30
different color all right all right so the formula I'm going
3:35
to put it right here right so is a x - X1 and then x - X2 right this is
3:44
the general formula X1 and X2 are given in this problem so here a is not given
3:51
so to make my life easy I'm going to use the value of a that is equal to one correct I'm going to use that right so
3:58
that's going to give me now x - -13 because this is X1 and then x - 2 all
4:07
right now I am not done yet I'm not done right I need to now develop this a
4:13
little bit I'm going to use a foiling method right so I have double negative so that becomes what positive right so I
4:21
get x + 1/3 and then x - 2 and now I can foil
4:27
right so yes sir two right two oh
4:33
six extra credit for the class that's not a mistake that's that's a big mistake how is that
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a big mistake confused how
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confused give him the half give him the half half x x right so X
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x² x * -6 that's -6x 1/3 * X that's +
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1/3 x and then 1/3 * -6 is -6 3
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right IUS I'm doing this because I want to put it in the standard form that's
5:13
right so that's why I'm doing it I want to put it back in a standard this is called a standard excuse
5:21
me I gave you the two Valu for X1 and X2 right we are going to the standard form
5:26
so we have to use this now this say root equal Z equal same thing excuse me does
5:32
it say root same thing yeah root Z that's the same way Sy yes when there's a number
5:38
outside of the parentheses do you just multiply the first parentheses or do you multiply both parentheses you do both
5:45
parenes and then you apply the number later okay so you you would multiply both parentheses by one by one yeah when
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I'm done yeah right when I get here I can multiply by one
5:57
yeah I did one * -6 so you foiled it right that's what I say foiling right
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I'm foiling this all right so now we can combine these two here right the common
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denominator will be three so 3 * 8 6 is 18 so it's going to be
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x² - 18 x over 3 common denominator plus 1/3 - 6 over 3 right this is two
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fractions I'm putting them together I shouldn't he no what here this is this is this is like pre-algebra Algebra 1
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can you make -6 3 -2 yeah and then I get x2 - 17 / X oh my gosh what's the number
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after X didn't you say 18 said 12 I'm combining like
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tering right adding be 19 that's 18 that's one oh oh
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you because I make mistake y'all think I'm just a pile of mistaking mistake making
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individual I don't know I I don't understand this right so this is the
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standard form all right now if I don't like if I don't want to have a fraction
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what can I do I can multiply everything by what three right so I get 3x 2 - 17 x
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- 6 so I I I suggest you do this to get rid of fractions right multiply
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everything by three to get rid of the fraction so what we just did here was was what we turn we were given the roots
7:32
and now we use those roots to solve to find the quadratic equation in it
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standard form does that make sense all right what doesn't make sense
7:47
I where x - 6 x I'm just multiplying this time this x * X is what x² right x
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* 6 is - 6 x 1 1/3 * X is 1/3 x and then
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1/3 * -6 is what -6 over 3 right then I
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combine the like terms here here this is 6X and this is 1 over 3 right I want to
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find the common denominator which is what three that means I'm going to multiply this one by what 3 so that get
8:22
-8x + 1 that is -7x and then 6 over 3 is
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2 right and then then here you multiply everything by what th three to get rid
8:34
of the fraction so we get 3x2 - 17x - 6 we good all right now move on to the
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next thing now how do we solve an equation uh by factoring the GCF we're
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going to solve a quadratic equation by factoring the GCF what what does GCF stand
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for okay so you know you're familiar with that right good all right so I now I have 16 X2 + 8 x = 0 I want to solve
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this quadratic equation so what do I do I need to factor that right because I can't just solve it like this so what's
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my GCF 8 what x 8 x is my GCF so I'm
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going to put 8X as my Factor right now I have two expressions left
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so how many * 8 x goes into 16 x² 2 what
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2x right 2x time because 8 x * 2x is what 16
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x² right so my common factor here is 8x right now I do the same thing here how
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much does 8 8X goes into 8X one so plus what one equal to zero
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right so this is how you factor the GCF out of an expression our
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greatest common factor is 8x right 8X goes into 16 x² 2X or if you want to ask
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yourself a question 8 x * what gives you 16x 2 * 2X or 8X * what gives me 8X 1
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what's the real life example of this we don't have any all right so now I'm not done yet
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right now when I get to this point I still need to solve for x that means I need to set each of this expression
10:23
equal to what zero separately right because if this product gives me zero
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that means either one of these is equal to Z so I'm going to go 8X is equal to 0 or 2x +
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1 is equal to zero and then solve for
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x now I'm going to divide this guy by 8 I get x equal to Z and then here I'm
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just going to solve it straight up right subtract the one I get 2x = -1 and then
11:00
/ by two it gives me x = -1/ 2 right so
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I have two solutions to this problem 0 and -1 /2 now I wanted to jump into
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perfect squares and differences of squares but I think I want to pause here for a second spend quality time on these
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two sections and then go to this tomorrow deal yeah all
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right uh why can't we just like all right so we doing part two of the
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section right yesterday we stopped at uh perfect squares and I did talk about it
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but I want to do it again because it's better I felt like a lot of you guys were probably just not even paying
11:43
attention so I want to do it again so we talked about how to solve equations we Ed what method yesterday remember I
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don't [Music]
12:00
fact GC GCF right the GCF stuff that we used yesterday okay now we're going to
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talk about perfect squares right so now if I have something like this x² + 16 x
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+ 64 we're going to use what we call the perfect square to solve it right now
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this is one of those identities this is called an identity this is always true right a s + 2 a + b s is equal to
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a square these two are equal right these two are the same thing so you use it
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this right here this is the factor form of this expression right if you have
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this expression you can Factor it this way all the time this is called an identity so
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basically ax² plus 2 A Plus b² right isal to a + b
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the whole thing Square this is true right this is called an identity right now how is this useful we're going to
13:05
use this to to factor this we have an expression here this is called a trinomial because you have three
13:12
Expressions right you have x² 16x and 64 so now our goal is to prove that the
13:19
first term is equal to a s the second term is equal to 2 a and the third term
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is equal to b² if that is the case then we can transform this into A + B sare
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right now first term is what x² so we know that x² is basically X the whole
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thing squar so in other words this is a squar right a the whole thing Square
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this is true so it checks out so we can check this off the bar right second we
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have this the third term 64 64 is basically 8 squ which is what b² so that
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also checks out right and then number three we have 16x so we
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we want to make sure that 16x is equal to 2 a so if you put two here a we
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already know is X and B is 8 so therefore 16 if you do 2 * X you get 2x
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2x * 8 is 16 x so that also checks out so since that's the case then we can
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rewrite this expression we can rewrite it as what as x + 8 the whole thing
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Square all right so you got to go through the steps first step is you want to make sure that the first term is
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equal to a the whole thing Square which is true second term 64 is 8 square that
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is b square that is true and number three 16x is equal to 2 a so if you do 2
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* x * 8 you get 16 x since that is true we have an identity all right so
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therefore we can Factor this and get this right here enter I don't think any the door is open
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I don't think anybody's outside oh you
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can all right so now we can Factor right we can
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Factor it so we have x + 8 2 that gives you x + 8 * x + 8 we can solve it we
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going to set it equal to0 and solve for x we get x = 8 okay all right now
15:25
there's another identity the other one is this a - b s is also equal to what is
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also equal to a 2 - 2 a + b s so how do we use this to solve this equation right
15:39
we're going to try to use the same thing to solve this equation I have 4 x² - 12x
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+ 9 right my goal is to transform this to look like this so what I want to do first is I want to make sure all my
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steps are covered right I'm going to start with 4x2 this is the first term right
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4x² can be written as what 2x the whole thing squ right because 2x * 2x is 4X s
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so that checks out this is a squ that checks out all right second we want to
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look at the third term is nine nine can be written as what 3 squ right so which
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is basically B squ right N9 can be written as 3 squ that is that is right
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now 2 a b step three is 12x can be written as what 2 *
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what 2x times no three right you always have to
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cheat right this is a and this is B right so this also checks out so all
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three check out hence you can write this as
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what as 2X X right and then now since -
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3 the whole thing Square equal to Z because everything checks out a square
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checks out b square checks out 2 a checked out so we can write this expression as 2x - 3 2 and then we can
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solve for x right so I'm going to set this equal to zero and I know that 2x is 3 and X is
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3/2 right this is called using the perfect square to solve this kind of
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problem all right all right you guys like that no I'm sorry you just have to
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do it right it's so confusing how do you do all that so I'm going to do it again
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all right so let's do another example let me do another
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one
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Bally yeah all right so let me um
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let me [Music] um let we got a problem here that that
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that's going require to use that and then we going to see if we can do it
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okay they don't really have a lot of the perfect squares here oh yeah all right
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let let me let me set this up here x² right what if I have x² watch this +
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14x + 49 and I want to solve for x right now the reason why here's the
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problem yeah what so the reason why right we will use the perfect squares is
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because the first the leading coefficient here is more than one right
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so it's easier it's easier to solve this we can solve this easily right I can
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solve this using what I can solve this using the the method that we guys have
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learned in albra one right I want two numbers whose product is what 49 that
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add up to 14 right two numbers whose product is 49
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by add up to 14 what are they s seven so therefore it's going to be what x + 7
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and what X+ 7 right basically right so
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here's what I suggest you do for this section if you come across a problem like this we're going to have to use a
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quadratic formula because I think this is confusing you you all right you can go back and watch it see if you can
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understand it but you can ultimately use this method here right which is when you
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have uh a second degree equation a tral like this you can use that method where
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you say I want two numbers which product is 49 add up to 14 is what 7 and seven
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and Factor you guys no this right so what if I have
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x² right + 6 x plus uh let's say 8 how
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would you factor this you want two numbers whose product is what excuse me you want two numbers
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whose product is eight that out of two what are they two and two and four so
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therefore this will look like what x + 4 and then x + 2 = Z and then you can
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solve for x correct so you going to have x + 4 is 0 or x + 2 is zero and then
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solve for x you get -4 you get x = -2 here all right you guys remember that
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right so this is the other way to factor trinomial now suppose that I have
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something like this let me give you another example
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here yeah
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um if I have this is probably the easiest one you going to meet or you going to come
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across right this is called difference of squares so what if I have I see if
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you guys remember x² = 64 and I want to solve this what do I do here um you just take Square 64 do
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you don't know square root yeah right we can square root both side right I'm going to square root this I'm going to
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square root this because these are both perfect squares right this is a perfect square this the perfect square so if I
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take the square root what do I get xal what not just8 what is it plus or minus
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8 it could be 8 or8 when you have a square 8 square is 64 and then 8 square
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is also what 64 so when you have this you take the square you're going to have plus or minus all right that make sense
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now what if I have let's say I have 3 y^2 = 27 and I want to solve for y what
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do I do first divide 27 by 3 divide by three
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right and I get y = 9 then what square root square root it right if I Square
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what do I get y = plus yal to plus orus 3 so this is basically doing the perfect
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squares to solve these equations you also have that option all right so now
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let me make it a little hard now so what if I have make it harder
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yeah we have 81 X2 - 9x is equal to Z
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right if you see something like this what is your first instinct we have to factor out the what
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the the G we want to find what what is the GCF here N 9 what 9x 9x
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right 9x I don't know so if I 9x what am I left with 9x S no 9x 9us what 1 equal
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to Z zero right so now I can set each one separately right I have 9x = what 0
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or 9x - 1 = zero and then solve for x
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all right so you have here what you have is this multip ways of solving this problem okay you're going to have
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problems that are going to require like the perfect square some problems are going to require something different but
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it's just for you to know which one to use at a time all right so now what I
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want to do is uh um um I want to work on some problems in
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the book and I want to see how we all going to affair on that right so if you go to um the homework doesn't exist on
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that page yes it does now watch this here right if you go to your textbook there's number five there
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and number five says 18 x² right we have 18
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x² minus uh - 3x there is no 1 through 10
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on page yeah it's only 11 through 25 of chapter 3 study guide what are you talking about on page 207 I'm on page
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242 you said 2077 who told you guys to glue on 207 you did I
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that's from my pre class jumping hand that's what when you don't listen to
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instruction 2422 yeah number8 on 242 right number
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eight2 number eight on 242 so now listen to the instructions
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there what are they asking us to do Factor each Pol they want us to factor each polom right they want us to factor
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these polom so what are we going to do we Factor how many terms do I have three
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four four terms so we have never come across such a fancy wait what are we on
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wa what number we on five I said number five no number five number five
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right right we not do number eight we doing number five we're going to do number eight when I'm done with this right so let's say we have um this poal
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I'm getting H so it's a fancy one8 here no wonder why so it's it's four terms how do you
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factor four terms polom can you combine the two you can you can simplify right
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you can simplify or better the the key here is this thank
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you to recognize some P right here's what I show you you can simplify but when you simplify it might be more
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complicated so the best way to do it is see if you can do a 2 by two right see if we can Factor this 2 by two uh we're
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going to we're going to do this here see it's the GCF here and G this is number five right on page 242
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if I look here right 18x - 3x what's my GCF between those two terms
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shake this two here what's my GCF 400 I don't
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know 18 three what if I put 3x what am I left with 6 6
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xus what one one now here what's my GCF here four four if I put out four what do
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I left with 6X 6x - what one now watch
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what do you see that is interesting here theide this is so now we have a
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common term right we could have a common term so we can put that common term as a factor right so I'm have now 6x - 1 *
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what 3x + + 4 right so when you have a four term
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polom what you laughing at something is funny right so that's how you do it here
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right when you have four terms you have to see if you can establish a relationship between two at a time right
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now we got this and that we found that our common term was what 3x see 6x - 1 right we find a
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common term so therefore we can pull it out as a factor and then leave be be done with like 3x + 4 does that make
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sense all right yes you good J all right so let's do another one here now I'm
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going to show you how to do another one here that that one is a little hard too won't it just come out to be the same
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thing if you do the other way if you just do 6 x -1 * 3x + 4 no all wait do
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you put the Z yeah you put up to zero and solve for x all right now look at this 2x + 7 x - 30 right I'm trying to
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factor this now this is not obvious right we're going to use a method called
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factoring by grouping right the grouping right so here's what I do here because a here is
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greater than one right the first thing I do is I'm going to multiply two now
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watch watch the step I'm going to multiply 2 and
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-30 all right you're going to multiply that and this so what's 2 *
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-30 60 right and now what I do is this I want two numbers whose product is -60
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but that add up to 7x right two numbers whose product is -60 but we add up to 7 let's think the
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the the product has to give you -60 but the addition or the sum has to give
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you negative uh POS 7 right
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yeah 12 - 5 excellent right 12 your shirt say
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12 12 * right gives you what 16 right well 12 - 5 gives you what
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7 wait right wait wait a minute let me finish this first so you see right so we're doing
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again the factor by grouping this is this is a really technical thing that you got to pay attention right so what
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you do is is before you solve it you always have to go you multiply the first term to a -30 you got -60 now your goal
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is to find two numbers whose product is -60 but whose sum is NE uh POS 7 so 12 -
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5 will be the perfect um combination right cuz 12 * 5 is60 and 12 - 5 is 7 so
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now what we do next is we are going to rewrite this expression and replace the middle term by 12 x - 5x so that we can
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Factor this all right so what we're going to do is this we're going to have 2x² and instead of 7 I'm going to
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replace the 7 x by 12x - 5x so that becomes now
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12x - 5x - 30 now I have the four terms
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polinomial and now I'm going to use the same method that I use here factoring by grouping right so I'm going to take
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these two together and these two together right now here between 2x2 and
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12x what's my GCF what's my
30:45
GCF 2x right if I put out 2x I'm going to be left what x
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+ six now here what's my GCF so I putus right I'm going to be
30:59
left with what x and then because it's a negative here I'm going to have right here a positive right because a negative
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* a positive gives me what a negative plus what six that's not 15 six right
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now I have a question who just wait a minute I'm not done so I'm done here
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right now you see well so now what's my common term
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what's my common ter X+ 6 what X+ 6 what
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2us what- 5 all right so do you see how we work this out hey hey you see that
31:43
the car yes you know okay all right so you got to learn how to do this all right so this is this is the section
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it's going to require fact Factor