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How To Use The Complete The Square To Solve Quadratic Equations
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Feb 19, 2025
In our last session, we discussed solving quadratic equations using the square root property. In the second part of this lecture, we'll talk about another technique, dreaded by many students- the completing the square method to solve quadratic equations. Chapters: 00:00 Introduction 01:13 Steps To Completing The Square 03:30 Solving an equation by completing the square 06:18 working on additional examples
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0:00
right
0:03
so I give you an equation here right
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it's a second degree and I say X2 + 16 x
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= 9 right so yesterday we we dealt with
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this kind of problem right so what did
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we do when we have this like X2 + 6 x +
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9 = 16 what did we do you take two
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numbers that multiply to get n and then
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two to add get six right and what was it
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in this case would be what three x + 3
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right yeah we had x + 3
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square and that was equal 16 and then
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what we do after that we square square
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root of it right now what the problem is
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there's a problem here what is what is
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missing here the third number the third
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number is missing as you can see right
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this is just x² + 16x equal 9 so the
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third number is missing that means we
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got to figure out how to get this uh
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what do we do this is why this comes
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into play or this is how this comes into
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play the complet the square so we have
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to figure something that we have to make
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some transformation because the third
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number is missing the third number is
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missing so that's how we're going to use
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this to solve this problem right so
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there are steps to completing the square
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all right
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so the first thing that we need to know
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is this the equation is given right do
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you have x² + BX right this is what we
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have here x² plus BX what do you think
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the value of B is
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in this case what is b b yeah b b would
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be 16 b b is 16 right this is x² + BX so
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this is the format that we have this is
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what we have here x² + BX so now what we
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want to do is we want to find the third
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number that needs to be added to this in
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order for us to be able to use the
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square root property that's what we're
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trying to figure out all right so now
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those are the steps that we have to take
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okay the first step is you're going to
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divide B by two yes sir that's still
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word in the third step it's add the the
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result right so step one you have to
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divide B by two that's just the step
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right so we have X2 + 16 we're going to
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go step by step what is B here uh 16 16
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so we're going to divide it by two what
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do we get eight eight we get eight right
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now the Second Step says what Square b/
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2 s so you going to square this number
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that you got right right so B over 2 is
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8 so we going to square that number what
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do we have now 64 64 and then the last
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step will be to add that number to the
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equation okay the polinomial that we
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have here so now in this case that will
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be adding what to it you'll be adding
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64 right that will be adding 64 to that
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right so now why do we do this why do we
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do this there's a reason why we do this
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and then the reason why we do it because
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we're going to use it to solve equations
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that are not easily solvable right now
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suppose I have X2 + 10 x - 11 = 0 and
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I'm asking you to solve this problem
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right the first thing you want to do you
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you want to isolate this portion right
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you want to isolate X2 + BX does that
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make sense you want to isolate that so
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that means the1 has to move to the other
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side correct so what we're going to do
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is we're going to have x 2 + 10 x = 11
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notice how I made some space here for
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purpose right cuz I'm trying to find
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what again jack I'm trying to find the B
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right the third number c pretty much we
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going to call that c we trying to find
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the third number right this is missing
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here so we're going to follow the same
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step what is the value of B here 10 10
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so we're going to do 10 and we going to
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divide it by what two we get what five
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and we going to square that 25 25 right
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but here's the thing when I add 25
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here I must add it on the other side as
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well all right you can't do one thing to
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one side of the equation and can't do
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and don't do the same thing on the other
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side meaning if you change something on
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one side you have to change it on the
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other side now we actually trying to
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solve this problem does that make sense
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right so now uh Nathan what do I get
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here I want two numbers whose product is
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25 x + 5 x + 5 S = to what uh 36 36 now
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can we solve this yes what do we do uh
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you sare both sides get x 5 square root
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both sides we get X+ 5 = what 6 or 36 6
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right or x + 5
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= -6 right and then we solve it plus or-
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plus or- yeah well I I already did it
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here it's it's basically plus or- 6 but
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I already separated right and then you
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solve it x + 5 will be = 6 and then x +
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5 will be = to -6 and then you solve it
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now now you've learned the meat of this
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section so now we're going to do some
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more problems to understand it because
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now it's not oh this is easy until we me
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some problem that can be problematic
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right so let's work on a few problems
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here and we're going to learn how to use
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this to solve this all problems problem
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don't erase no I'm not going to erase it
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no I'm not so suppose I have uh
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suppos I have
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[Music]
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um all right one one of them can I erase
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this yeah just don't erase the right
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side all right actually I want to stick
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leave the steps because you might forget
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the steps can I raas this though no this
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no no you can raas the the first yeah
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yeah yeah let's raise this here head
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away no all right so now blind suppose I
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have x²
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right uh -
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4x + 12 is equal to zero what is the
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first step that I that I take subtract
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12 subtract 12 right but before you do
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that always double check to see if this
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can be factored can it be factored here
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absolutely not right there's no number
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that m 12 that add up to what4 so we
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can't do that so we have to subtract 12
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right so now we're going to get x2 - 4x
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+ that = -12 + Dash right the dash is
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the number that is missing does that
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make
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sense yes or no yeah sure all right so
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now what is B we say no what's I they
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wrong but the 12 was already there so
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well can you fact is that two numbers
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that multiply to 12 that add up to4
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no huh no yeah actually we could have
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done it Go 6 and two right 6 and -2
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actually no it's not going to work it's
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not going to work because 6 * 2 gives
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you 12 or
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6- the same number H doesn't it need to
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be the same number no but it's not going
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to work because 6 * -2 is -12 and then
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-2 * 6 is -12 we want it to be 12 so
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that's not going to work all right
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Square to be the 12 yeah right so that's
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not goingon to work so we have to use
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the completing the square so we subtract
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the -12 right now we need to find the
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third number
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here and how do we find that we're going
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to take the -4 here right4 we going to
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divide it by what two that gives us what
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and we going to so what's -2 squ so that
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means I'm going to add four four here
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and four here right are we good now
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Nathan what is this going to give me I
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want two numbers whose product is x - 2
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x - 2 2 that's it =
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to8
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now we got a problem here right so look
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we have
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A8 right x - 2 imaginary numbers in here
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so we're going to use what imaginary
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numbers there we go right so we going to
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have x - 2 equals what plus or minus
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square root of 8 I right8 yeah8 which is
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fine and now we got to use what the I
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right because it's a
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negative right it's a negative so now we
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going to have x - 2 equals to what8 will
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be plus or minus I < TK of 8 and now we
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have to break down the squ root of
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8un of 8 is what 4 * 2 which is 2 < TK
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of 2 so now we have x - 2 = plus orus 2
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IUN 2 and now we can solve it right so
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we're going to have x -
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2al 2
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i x - 2 = -2 I actually that was funny
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so I have to laugh when you get ZX right
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when you get Z no or that means it's or
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oh I didn't right and then we can solve
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it so if you plus it would it just be
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two + 2 I so X will be 2 + 2 I < TK 2 or
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X will be 2 - 2 I < TK 2 right so this
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is dealing
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with yeah I
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don't that's that's fine everything that
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we learn new is always like that like
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hard in the beginning but then you how
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would I check that like I were to
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substitute it back in would it be
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possible for it to be equal to each
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other so many steps that's not how many
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steps now let's do another one would be
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using there's like 16 different
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equations that start with
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x dangling he's dangling on that all
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right let's do this
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one all right so x + 8 plus 10 equal to
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zero somebody a benevolent person walk
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me through the process please yeah go
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ahead you want to do
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it yes sir what's up how you doing
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I'm can we help you you're born come
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join us
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no no leave why you
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can all so what do we do here 10 minus
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10 thank you very much wa what I want to
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yeah you should solve the equation right
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no d
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= right so 8
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2
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4 doesn't matter the situation goes back
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to x +
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[Music]
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4al what is that is it just six six okay
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and
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then can you square or no
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6 just of 6
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so 6 so what would you get then x + 4 =
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plus
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or and then X will be now there's a way
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you can write this without doing all
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that stuff you can just go - 4 plus Aus
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6 and you can leave it like this you can
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do that too okay that you can write your
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answers like you don't have to split
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that you just put4 that's minus 6 if you
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want all right I would like to do that
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okay yeah
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abouts all right so this is how we're
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going to do this now tomorrow what I
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want to do actually no I want to do on
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Thursday we're going to talk about
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equations that involve a number here
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that is not equal to eight so now I'm
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going to get you some some practice
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problem to work on yes sir yeah all
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right and then you should do this give
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Ben some the questions too we don't have