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Factoring Quadratic equations , Perfect Squares, Factoring By grouping And More
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Feb 19, 2025
Welcome back to our class , today we’ll spend a significant amount of time on working on different wrinkles related to quadratic equations. We’ll deal with factoring trinomials, factoring binomials , factoring by grouping , factoring perfect squares , and then using all these techniques to solve equations.
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Jing that's why I'm just joking now can't pay ATT to you want to find what we want to
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write a quadratic equation in the standard form with the given root right
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remember so all we have to do is just set up the problem so a quadr didn't ask for the so we can choose a to be what
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one to make our life easy right so we're going to have 1 * x - -5 right and then
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X - 12 Mr do you have another one of these is yeah right so now we have x +
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5 and x -2 now one of the things that I've seen is like when people are
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foiling this they forget into like times it by two at the end or the denominator right so it's going to be x and x that
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is x² x and2 is2 x and then 5 and X is
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5X and then 5 and2 is 52 right so now what I want to do is this I want to
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multiply everything by two so I can get rid of what the fraction the fraction right so I'm going to have here
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2x^2 - x + 10 x - 5 and now I can just combine these two here it gives me
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2x2 + 9x - 5 so that's the goal is to do that you always want to make sure you
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have uh integers instead of fractions where' you get a from a that's nine oh
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that's that's not an a that's a nine right nine where you get a G from that's
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nine that's nine that's not a g it's not a a where you get the P from what about the T there's a t that's a plus sign
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cool where'd you get the L quiet what so then the next thing we're going to do is we're going to learn how to factor right
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and some of the problem you guys some of you guys were doing too much you have to read the instructions it's
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saying Factor it didn't say solve it right so if I have
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40 a s - 32 a the question is asking me to
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factor they didn't say solve it they just want me to put this as a factor so we want to factor this so what am I
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asking what what am I asked to do GCF find the GCF so what's the GCF between
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40 a squ and 32A eight8 what uh 8 a 8 a and all I have left is
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what 5 a minus four right that's it some of you
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guys were like solving this to zero I'm like this is too much work they didn't ask you to solve they just want you to
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factor it right so stick to the question Factor so we have to read better right
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now what if I have 51 I have another one 51 C Cub minus uh 34 C C what right 51 C
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cubus 34 C what's my here what's my
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GCF C C is good you start with C why the numbers why goes into 51 that goes into
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34 three how do you know if number goes into like wait yeah it is it is three
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three doesn't 34 I'm trying 17 17
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17 goes into 51 three times three times it goes into 34 twice how did you figure that out well divide 51 by 3 you get 17
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so you just kind of reverse 1 here's the thing if you can't if you want how do
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you know the number is divisible by three add the digit right 3 + 4 is 7 is s divisible by 3 no so therefore it's
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not going to work but I know that two goes into 34 but two certainly doesn't go to 51 but what two goes into 34 I
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have 17 left so I'm going to test it it's just by testing I have a question what is it
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um for here am I going up or down or I can't tell you what it's inside so what
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is it up or down no okay thank you so what's that 3 c^ 2
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minus what two two right and we're done because once you get to this point
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there's nothing that goes into 3 c² that goes into two right so you stop right
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here I'm I'm just speaking those that I feel like will give you a hard time and and we going to work on those now 22s is
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a little bit different CU 22 has a lot right we on we on page uh 242 so let me
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let me write down 32 32 has 32 I mean
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22 all right now if you look here right I have turn terms do I have four four
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terms right so whenever I have four terms that means I'm going to do them 2 by two right so I have four terms here
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so I'm going to look right I'm going to look at these two together and I'm going to look at these two together to begin
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with right so if I look at 32x Y and 40 BX what's my GCF 8 8 what 8 um x x so if
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I PL 8 x what am I left with is it four y and 5 b plus 5 b right
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remember I'm always respecting my signs as well now what about 12 a y and 15 a
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what's my GCF or is it negative it could be3 what
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a a notice what's going to happen What when I take out3 as my GCF what am I
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left with here 4 y 4 y 4 Y and then plus plus 5B you see it's always going to
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happen and now what's my new GCF 8x 3 a right so this is my now I have what 4 y
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+ 5 b parentheses and what 8x - 3 - 3 a
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right so this is how you factor this so do you see whenever you have four terms always try to see if you can Factor them
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by grouping right this is called factoring by grouping because I want to put these two together these two
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together and see if I can find my GCF and then proceed that way all right so let me find another one that I believe
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might be problematic do we have a com up no not not till like we come back I'm not doing the test again right now thank
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you yeah I don't get Christmas if I can all
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right now let's check this one out here right I 15 Y 2 right 15 Y 2 -
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240 right 15 y s -40 what do you think I'm going to do
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here what will be by GCF is it let's
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see does 15 goes into 240 I don't know let's try does 15 go to 240 yeah how
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many times all right so if I put 15 my GCF what do I have left y Square minus
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what 16 right am I done that's where the question am I done
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here no I'm not done no the Y is a square Y is a square right now look
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here you guys know about the difference of two square we discussed it right a square - b square is equal to
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what no no Aus B * what a plus b a plus
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b we're going to apply that here this is called the difference of two squares right two perfect squares right so I
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have a perfect square here and I have another a perfect square so what is that going to give me now is it 15 15 y uhhuh
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wait no it's going to be what if I apply this a b s a - b a + b y - 8 y - not 8 y
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- what 16 4 oh four right cuz four is a perfect square right and then what y + 4
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right are you done M all you can put it on a des so y - 4 y + 4 right you guys
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got to remember that this is the difference of two perfect squares okay this gives a lot of people the hard ATT
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because they can't recognize you have to be able to recognize the identities all right let me give you another example what if I have uh let's say I have this
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here so how would you factor this 3x2 - 27 first what would you pull out three
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three right if you pull out a three you be left with what x²
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- is it8 no nine n and then what what do you are you done no we have what the
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difference of two perfect squares right so that gives us what 3 x
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- 3 x + x + 3 all right so we got to remember the difference of two perfect
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squares this is very important okay and then we stop here we're done
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for this one right now let's keep moving um another one that I I feel like
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we should do is uh let's say 27 I have
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X2 - 9x - 22 now this is a trinomial
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right I have how many terms three three terms when I have three terms I'm always thinking in terms of all right now I
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have three terms my leading coefficient is one right is one here because there's
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no so what I'm going to do is this I want to find two numbers whose product is what -22 but who Sol is 9 right find
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two numbers whose product is -22 right we want the product let's put P here and
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ask for some 11 and two right so I know that1 and two is going to give me what n
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n no * 2 -22 right so but since I want this to be negative there it's going to
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be1 right so it's -1 * 2 and then1 + 2
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gives me what 9 so that works so because that works now I can Factor this uh
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trinomial now I got what x - 11 and then
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x + 2 so do you see how we work these things out you want two numbers whose product is the last one which is C and
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then which Su gives you what the middle number which is9 okay and then you factor them out you give a practice
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problem yeah all right so for practice you guys can do uh write it down I don't
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yeah x² right + 13x + 40 how would you do that you
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want you I want you to factor it two numbers whose product is what 40 the product has to give you 40
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and the sum has to give you is it this right 8 * 5 yeah so 8 * 5
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so this will be therefore Factor will be what x + 5 and then 8 x + 8 right
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because 8 * 5 give you 40 8 + 5 give you 13 so to factor it is x + 5 and x + 8 so
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when the leading coefficient is one is super easy to do but then now we're going to turn it and if you look at for
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example if you look at number number 28 right there's a little twist there we have 3x² right that's number 28 +
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12x - 36 before we solve it right what do you think you want to do if fact if
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I'm looking here I'm looking at my leading coefficient is what three so I'm
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like I this is not as obvious so first I want to try to pull out the three out of that right as a GCF cuz three goes into
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three three goes into 12 and three goes into 36 so first you can pull it out as
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a GCF so now you get what x² + 4x minus what 12 now can I work
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with this yes right yeah so now I want to find two numbers who some which product
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is -12 but then whose sum gives me four you see what here right at first
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you put out the three as a factor yeah and six and two right but since I want
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this to be positive then which one has to be negative six six has to be what positive
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positive there you go and then two so therefore you factor is x + 6 x - 2 that
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make sense first we pull out the GCF to to make sure we have a line coefficient
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of one if we can if we don't if we you can't do that we have a different technique that we have to use which
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we're going to see next right all right now let's look at for example 29 you see
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you're not going to like 29 29 says 15 x² right + 7 x - what 2 okay this is now
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a different technique that we're going to use here we're going to use a different technique because if you look here right what's my leading coefficient
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15 15 can I pull it out 15 no I cannot do that so now the Techni that we going
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to use now is this you're going to multiply the coefficient this number 15 *
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-2 right we multiply it 15 * -2 that gives you -30 right and now we proceed
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the same way we want two numbers whose product is -30 that we add up to seven
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Sol has to give me seven the product has to give me -30 is it two let's see can you do a negative like
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can we find a number like that cuz what think about there is the
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combination that will do that if you look at 30 right 10 and three right so which one has to be
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positive 10 is positive 3 is negative so it's 10 and3 right 10 - 3 is 7 right and this is -30
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so now the next step is crucial so the next step is you're going to rewrite
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this whole expression by replacing 7x by the middle number that we got
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let me say that again you're going to rewrite this whole expression by replacing the middle number by the two
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digits that we found right so we're going to go back and say this is 15 X2 + 10 x - 3x -
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2 you see what we did here so you rewrite the whole expression and now since we have four terms we're going to
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use the grouping again right now we can do these two together and we can do
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these two together all right so what is the GCF between 15 x² and 10 x 5x so 3x + 2 3x +
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2 right and here if I don't have anything what i p out always negative 1
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right here be 1 if youly 1 you get what 3x + 2 do you see how this works out
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perfect and now we have our GCF so the whole thing is us what 3x + 2 * 5x
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-1 do that make sense so it's a lot of a lot of things that we are learning here like a lot of different techniques that
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we use so I'm sure you guys did some of it in algebra one yeah what is the point of getting that though getting what uh
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the factor like parentheses yeah oh it's let's say if you wanted to solve this if
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you have this is equal to Z you want to solve for x now you can go ahead and do
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what set each one to zero and then solve for x all right so that's why we Factor
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first you factor and now you can solve it if you don't know how to factor is hard to solve it you have to use what
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the quadratic formula which is a pain in the neck I don't like the quadratic formula it's not it's not hard it's just
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a lot of things and I don't like using it I just like factoring right so now
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let's work on 30 30 let's work on 30 first and then
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we're going to work on 32 if you want right let's do 30 30 say what
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4x² right + 29 x + 30 you guys want to
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use work or not as yes right 4x2 + what
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what is it 29 x + 30 yeah all right so what do you do first do you have to
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times it times it yeah you times what what's 30 * 4 *
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120 now you want two numbers who product is 120 but sum is what
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29 so now you have to play with it and if it doesn't work that mean
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sometimes it's it's not factorable right but you can kind of play with
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it anybody a number so 30 6 and five and four right
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wait 30 I thought it's 120 24 and 5 yeah all right good dude how
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did you do that so 24 and 5 right 24 + 5 is 29 you see you got it so you just you
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just have to play with the num right cuz I broke down I was breaking this down so I can see if I can come up with a
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combination 2 and 2 is 4 4 and 6 is 24 24 + 5 is 9 right you just have to play with the
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numbers it takes sometime it takes a little like 5 minutes to get to it so now I'm going to rewrite this as 4X s +
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24x right + 5x + 30 and now we can do the grouping here right so what's my GCF
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between 4x2 and 24 4X right four that be
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what x + 6 right and here we're going to take out 5 x + 6 and now we have the
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same thing so we have x + 6 and then 4 x +
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5 and then we done all right okay can you let us do the next one by ourself yeah I'll let you do the next one by
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yourself can you write it down 18 x² 18 x
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2 + 15 x - 2 you go ahead and do that
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byself no - 12 not 2 - 12 18x 2 + 15x -
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12
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wait
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it did this not feel like working today you time
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anybody got it yet all right you have to times the 18 right m is there any like
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is there any solution to like finding what numbers in like do these um what you get
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for 18 * -2 216 yeah all right here's what you
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can do right so you can put 18 and -12 here and then what you can do is break
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down 18 18 is like 9 and two right and 9 is three and three and 12 is like three
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and four and four is 2 and two so now find this combination here that can give you 15 right for example if I do uh
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let's see six here right six and uh this is two and three six maybe I'll do two
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two eight yeah if I do 8 and 27 27 - 8 is what 15 you see cuz you do
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that look I got three here right three and three 3 * 3 is 9 9 * 3 is
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27 right but how do you know to pull out from the nine and not from the two I'm just playing with the numbers I just
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have to play with the numbers to find a combination right cuz I know three and 3 is 9 9 and 3 is 27 2 * 2 * 2 that's 8 so
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that's 27 * 8 right and I know that's going to give me whatever the number I'm looking
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for but 27 - 8 also gives me 15 so that would be a good candidate do you see
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what I'm saying so you just have to break down the numbers and then play with them until you find that combination there's no like one way of
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doing you just have to figure out what combination gives me there all right and then you can Factor so we're not going
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to work on this cuz you know what to do from here right can you give me another one so I can do your method and try all right uh let's see
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uh all right let's do this one 12 x²
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right plus 13x - 14 try that 12x2 + 13x
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-
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14 yeah can you pass me one two
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see all right so you already got
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him don't know all right so let's see all you broke it down good right 13 no
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13 is just what no no no
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129 24 so's see so we have 2 and seven here right okay so now if you we want 13
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as our combination let's try nine and then 2 and two is what 4 and 7 28 28 and
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N that doesn't work right now what about let's try now 2 2 is 4 4 and 3 is what
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12 12 and 21 that doesn't work either right so what if you
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[Music] do 7 and three no we did that
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two two two four right four and
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uh oh no this is three and two oh my gosh you're giving us a headache right
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so two and two and two that's what eight8 and 21 let's see 8 and 21 21 21 and8 yeah
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that's worse yes you see so here I don't
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understand how you get that I don't it's no magic I'm just playing with the numbers do you see you just have to play
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with it it's not going to come like overnight he just playing with number yeah so have to find something that you
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multiply these two first right and break them down you always have to like here's
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the thing you want these two first right 12 and 14 or -4 right and you want to
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break down this here right 2 and three and this is 2 and 7 so you have one two
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so let's write the numbers down we have two two 2 3 and 7 we want a combination
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here that's going to give us 13 right that's what we looking for so
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if I'm looking here right 3 * 7 that's 21 2 * 2 * that's 8 right 21 - 8 is 13
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so that works you just have to play with the combination there's no like right you just have to be able to
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play with the numbers to get 13 in the middle that you see what I'm saying so it's just a matter of like doing a lot more of these and you get comfortable
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with it got something wrong here yeah of course when you pull out a -2 right this
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should be negative cuz a negative and negative gives you positive right no is that oh no it's negative up there yeah
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so she pull out the yeah so this should be this should be positive here this should be positive
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look y got it right so we have what 21 and 8 so
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replace 13 by 21 - 8 and that should work all right 21 and8 so that should
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work you do another one yeah is this it is is there One More
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Concept you teach uh there's no other concept we've learned pretty much every single one of them but I do want to work
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on 33 first okay let me do 33 right we have 9x2 - 25 and I want to factor this so
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what does that look like to you look at that for a second guys 9x2 - 25 * 9 what
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does that look like so is it a² a square - B squ right
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the difference two perfect square so what's the perfect square here 3 3x
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right the whole thing squar and what's the perfect square here or five 5 squar
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right so this is like your a and this is your B you see what I'm doing here I'm breaking them down so it make sense
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right so this is your a the whole thing squar B whole thing squar so therefore
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it gives you what 3x - 5 and what 3x + 5
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right you have to be able to recognize these are the easiest one when you have like difference of two perfect squares are you going to tell us on the midterm
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if it's going to be when are the I might January 17th or 15 right I
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might are we going to start studying for them soon yeah we going to do that soon are we going to do that after break or
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yeah after the break can we just take like the whole week to yeah we we probably yeah it's going to be one week
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right but can you give it to us on the Friday so we can work on it in the weekend and week after yeah all for
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example look at this problem 15 x² is ugly but we going to work on it right -
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84 x - 36 = 0 you remember when Cole was saying how when are we going to use this
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now we can use this here right I want to solve this so to be able to solve this I have to be able to do what factor that
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right so if I look here right I want two numbers whose products is what this and
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this and sum gives me what 84 so I'm going to break down I'm going to put 15
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and - 36 here right 15 is what three and five and then 36 is 3 and 12
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right and this is like four and three and this is two and two so I have how
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many threes do I have let me lay them out I have two two three three three and
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then five now I want a combination that's going to give me 84 it's pretty ugly right so let's see let's see what
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we going to do here right I want to combination that giv me four 3 * 3 is 9
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9 * 3 is 27 right so if I have 27 * uh 8
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* * 40 that doesn't give me anything 40 where'd you get 40 from 4 and 20 four
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and 5 is 20 that's no good wait don't you multiply 15 and3 yeah you do 54 4 yeah and then you want a
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combination from that that give you 84 right so you see I'm not even I I
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have to look at it now and play with the numbers until I get that my your Way's
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confusing yeah you just have to play with them right you play with them and you get to that point so somebody does
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it so enlighten me on this so let's see what if you do 2 * 3
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that's six 6 and 5 is 30 30 and uh 18 that's no that doesn't work 906
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what did you get M what did you get 906 90 and6 yeah thank you 90 and6 wait
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wouldn't it be90 + 6 yeah 90 + 6 yeah good thank you
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Jack all right so now we have what 15 X2 right - 90x + 6 x - 36 is = 0 again on
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the test you going to have OV time to get this done so do not worry can you give us but nobody Chim in CU I want to
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do it want
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camera right so now we can do these two together and these two together and then you guys can after that you can Factor
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it all right this it so 15 so 15 x here
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so that's going to be x minus what 6 right and then here I'm going to pull out the 6 I get x - 6 = to 0 now I can
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solve this by pulling out my common factor x - 6 * 15 x + 6 = 0 have X = to
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6 or 15 x + 6 is zero and then 15 x is 6 and X is 6
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over 15 and you can simplify that to 2 over five right and you have your two
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solutions here so see this is how we do this kind of stuff work all right we
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have like a couple of more minute left so I'll do two more problems and then we can stop now how we going to do this
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here check this one out x² = 121 how do I solve this
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problem x² = 121 what do I do here did you grade our test yes I
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already give it back to you I didn't see oh I give it to you today did I get a bad grade on no 11 what I do yet plus or
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minus 11 right when you have this stuff you take the square root of both side
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and X is what plus or- 11 is 11 or1 cuz it could be either 11 or1 cuz1 s is 121
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and 11 s is 121 so remember that don't forget all right can you give me another one of those problems yes I can so we
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can stop here for now right and then we're going to continue on
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