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How to Solve Quadratic Equations By Factoring
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Feb 19, 2025
Quadratic equations
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0:00
so we we talk about section 3.33 which
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is basically solving quadratic equations
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by factoring right I do not want to
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spend too much time on Section 4.1 or
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two because it's just graphing and to
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solve a quadratic equation when you
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graph you just have to find a point of
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intersection in this case we're not
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going to do that we actually going to do
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it analytically right so the first thing
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we need to understand is what is the
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standard form and what is the factor
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form you guys heard about the standard
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form of an equation of a second degree
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right now if you look here this is
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called the standard form any equation in
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the form a
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x² + BX + C is in a standard form and
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this is a quadratic equation why is it
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quadratic because it makes a Quadra
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no that's the why it's quadratic because
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it's a second degree equation right well
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I know quadratic s like quadral words
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now quadratic means it's a second degree
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what's a second degree polinomial what
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is
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it yeah four second degree meaning the
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highest degree is two so it's a second
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degree if it's a third degree is a cubic
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function ax CU plus whatever this is a
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second degree mean you see Mr salami yes
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hi do you have Sabrina she just went
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through the restroom to expl okay when
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she comes back can you um have her come
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to the front office for early dismiss I
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shall thank you thank
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you all right so ax² + BX plus C that's
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called a standard form does that make
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sense right so I give you an example
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here we have X2 - 8 x + 12 so if you see
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any equation in this form this is called
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a standard form right now the next layer
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is called the factored form which is
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self-explanatory right factor form
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meaning is it's like this it's as a
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factor Okay so so we have a * x - X1 * x
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- X2 the example is x - 6 * x - 2 what's
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the value of a in this case one right a
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is one so it could be anything but in
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this case we chose to put a = 1 okay now
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one of the questions that people are
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often encoun and struggle with is this
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type of question I say write an equation
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a quadratic equation with roots 1/3 and
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6 do you even understand this question
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maybe okay what is is saying is this
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they want you to build the
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equation and we know something we know
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that the equation has two Roots meaning
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1/3 and
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six that means that the equation is
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actually
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crossing the x axis at some point right
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so because the equation has two Roots we
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can build the equation using the factor
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form
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does that make sense I know the roots
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right that mean the equation the
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equation is crossing or the the function
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is crossing the X AIS at two Roots do
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that make sense am I making sense here
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right we have two the function is
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crossing the the x- axis at two places
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you've been called all right to the
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red all right so basically what we're
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going to do is we're going to build that
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equation so the equation is going to
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look like this now they say write a
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quadratic didn't say write B quadratic
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we don't obviously know the value of a
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so we can call a one I chose to let a
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equal 1 so I'm going to start by saying
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this is equal to a * x - 6 and then x -
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-13 uh do you see what I'm doing here
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I'm trying to build the function I've
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been given two Roots meaning these are
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the two solutions to this equation I can
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actually build a function based on the
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roots right so now to make my job easier
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didn't ask me for the equation they
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write they ask me for an equation so I'm
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going to let a equal 1 to make my life
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easier cuz I don't want to do too much
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right so now I have x -
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6 * x +
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1/3 and this is the factor form I can
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stop here and say I have found my
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equation no one asked me for standard
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form they just say write a quadratic
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equation I can stop here yes what do we
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do
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with quadratic equation say it again
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what if I ask for the equation that I'm
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going to give you I'm going to let you
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know that a is equal to one or a is
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equal to two I'll give you specific
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information it it won't just be like
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that yeah does that matter which spot
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six is in which spot okay it doesn't
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matter because it's commuity it's
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multiplication right so now if you want
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to go step further if I was asked you to
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find this in the standard form to find
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this in the standard form you have to do
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what foil this whole thing multip
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multiply so you have to do to do that to
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the standard form you have to for so x *
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x x * 1/3 -6 * x -6 * 1/3 so that gives
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you pretty much
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x² + 3 x - 6 x and then - 6 * 13 is two
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and you can go ahead and combine like
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terms and this will be called the
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standard form does that make sense so
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basically this is just an introduction
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we just need to know what is the
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standard form and what is the factor
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form now we're going to learn how to
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solve quadratic equations right now if
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you look at this quadratic equation here
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or what what do you see that is going to
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add value like I have 16 x + 8x = 0 what
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is what is interesting about this
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function here of this
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equation AAL Z zero right
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usually standard form is ax² + BX plus C
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in this case C is equal to
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what0 right I just have two terms I have
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16x2 + 8x = 0 so how would you solve
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this equation how would you do
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this Jack how would you solve this
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equation we factor out
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what what is this called greatest common
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factor the greatest common factor right
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to solve this have to factor
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so what's my
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GCF the GCF is not eight it's not just
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eight you have eight what 8X right the
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GCF is 8x so I'm going to put 8X as a
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factor if you did algebra one we discuss
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this right so I'm going to put 8X as a
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factor so if I put 8X what am I left
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with for first
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time
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2x right plus what
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one so you see what I did here right you
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put 8 no one because 8X * 1 is what 8X
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yeah right so when you factor out the
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GCF you have to think in terms of
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division right so the greatest common
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factor of 16 x² and 8 x is 8 x does that
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make sense right so if you put 8X as a
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factor you're going to go what * 8X
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gives me 16x2 2x obviously right and
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then what * 8X gives me
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8x1 and now you can solve this equation
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by setting each one of these separately
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equal to zero right so you're going to
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go
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8X is equal to Z or 2x + 1 = 0 and then
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solve for x does that make
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sense we have a product right we have to
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separate them and then solve them
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individually are we good yeah so that
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means we divide by
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8 so X is zero
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or 2x = -1 and x = 12 so your Solutions
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are X = 0 and then x = 12 this is how
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you solve a second Dee equation with Cal
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Z by using the
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GCF do that make sense now the next
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thing is using the perfect squares to
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solve this right now before we talk
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about perfect squares I need to explain
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this equation I don't think some of you
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guys have done it
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so this is called an identity a + b s
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somebody did some math several years ago
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and realize a + b s is = to a
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2 + 2
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A plus b s this is called an
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identity identity did you do you did you
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know about this identity I learned about
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this yes this is called an identity now
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how do you use the identity to solve a
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quadratic equation is what we're going
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to learn now if you look up here I have
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an equation
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x² +
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16x + 64 is equal to Z so how do I go
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from that to this is my job right so
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what I need to find is I need to find
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two perfect squares first and I need to
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double check to see if the middle
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numbers multiply together will give me 2
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a so if you look here right if you look
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at the extremities I have x² and 64 is
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x² perfect squ yes it is right I have X
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squ so I'm going to call x squ a squ and
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then 64 is also a perfect square 64 is
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what is 8 squ right so now this is
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basically X parenthesis squ = 8
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Square now I can go from here and then
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build uh
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this okay I can build this from here
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because I know that x² is pretty much x
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square which is a square and then 64 is
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8 s right so then I just have to double
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check that two a 2 * X and 8 gives me 16
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x let's see if I the 2 * X x * 8 which
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is basically 2 * a * B I get 16 x so
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since this is true therefore I can
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rewrite this equation as x + 8 2 = 0
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basically I'm using my perfect square to
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solve this
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problem you get
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it you have the quadratic equation you
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want to see if this is a perfect square
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yes wait I'm a little confused cuz isn't
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16 also perfect square only need two
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well 16 is in the middle you want you
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always want to start with the
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extremities like x² and 64 you reorder
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it because it's a x s + B X plus C right
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so what you want to do is this you want
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to verify that 16x can be a perfect
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square because you have 16 x there an X
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here so it's no longer a perfect square
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16 is a perfect square well this is the
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whole expression is 16x right so now
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here's the problem sometimes it's not
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quite evident to do this so if you feel
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like this is too much just use your old
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the old method I want two numbers whose
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product is what 64 and sum is 16 what is
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it you want two numbers whose product is
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64 and sum is 16 x let's do that I have
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X2 + 16 x by 64 I want two numbers whose
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product is what
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64 and so is 16x what is it 8 8 right so
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you just go x + 8 and what x + 8 no
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minus x + 8 and x + 8 wait so how when
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do you get the X Plus xus well you going
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to get that if you depends on the
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numbers here you have a positive so if
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it's positive positive and this is
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positive you want two numbers with
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product is 64 and S 16 x so it's 8 and 8
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does that make
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sense are we good
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are we good on that all right and now we
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can just solve it so I have now x + 8 *
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x + 8 = 0 how do I solve it I just split
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them because it's the same expression
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I'm just going to have x +
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8 is zero and then subtract 8 on both
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side we get x = 8 since it's the same
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expression that has been repeated twice
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we don't have to solve it twice because
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it's it's going to be the same
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answer okay
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okay confus confused why wouldn't you
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just do foil foil foil what do you
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mean like here I'm trying to solve for x
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yeah never mind right let me give you
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another example let's work on another
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one you still got to divide it to one
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let's let's work on another one real
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quick you see let me let me put this
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here so what if I
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have
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x²
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um
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all = 0 how would you solve this problem
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problem x² + 6 x + 9 = 0 what would you
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do
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here
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X+
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M square right so be x + 3 S which yeah
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you got that x + 3 S because they
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you got it or let's say you didn't know
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how to do use that method you just go
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okay I want two numbers whose product is
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what 9 and some is six so what x + 3 x +
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3 yes sir so will always be equal zero
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or yeah because that's what we solving
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okay I just this an equation I gave you
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so I want it to be equal to Z so we go3
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X will be3 so X will Beal all right now
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let's try another
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one so how would you solve is X2 - 10 x
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+ 25 = is it is the first one going to
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be subtraction so it's going be x - 5 x
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+ 5 no x - 5 x - 5 but the second one's
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positive it don't matter yeah so just if
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one's negative they're both negative
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well if one is negative here because
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basically what you want to do you want
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two numbers product is 25 and some is
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-10 right so it's five five but since
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this is -10 they both have to netive cuz
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negative time okay okay okay okay got it
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so then your thing is x = 5 yeah x - 5 x
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- 5 or x - 5 2 if you want right and
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then solve it so X is 5 does that make
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sense are we good all right so I think
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this is it for this thing now we can do
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some problems and figure it out go