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How To Solve Second Degree Equations With Rational & Irrational Roots With The Square Root Property
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Feb 19, 2025
This lesson is part of a two parts lecture that mainly deals with solving second degrees equations by completing the square. But before we get into the meat of the matter, we first need to learn how to solve equations with rational and irrational roots. Chapters: 00:00 Introduction 00:43 Using The Square root property ( Equations with rational roots) 05:48 Using The Square root property ( Equations with irrational roots)
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0:00
what this section is going to deal with
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like solving um equation with rational
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roots and then irrational Roots but
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we're going to talk about like
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completing the square but before we do
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that we first have to talk about the
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square root property and that the square
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Ro property is going to Encompass two
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things which I just mentioned dealing
0:20
with equations with rational Roots which
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is the one that have like perfect square
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where you can solve it right away and
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then we also going to talk about the one
0:28
that are irrational where you be like s
0:30
root of 27 and then you have to still
0:33
solve it but then your answer is going
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to be irrational
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right so now uh the very first example
0:40
that we're dealing with here is this one
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here right this is a second degree
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equation I have
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x² + 6 x + 9 = 36 and I'm trying to
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solve this right so the way you do this
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is this most people are going to want to
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like uh subtract the nine and then deal
0:59
with it but that's that's not going to
1:00
help right so the first thing you want
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to do is you want to
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factor this side right you want to
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factor this and then the way you factor
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this is very simple we've done it before
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right so I have X2 + 6 x + 9 so my goal
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is to do what I want two numbers whose
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product is what
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nine but then we add up to like six what
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are the numbers three and three three
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and three right so this gives
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you a x + 3² = what 36 right x + 3² isal
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to 36 that's what we
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have okay now I'm trying to solve for x
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so what do you think I'm going to do now
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I want to solve this for x x = x = what
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so what would you do I have x + 3 the
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whole thing squal to 36 and I'm trying
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to solve for
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x somebody say Square who say that
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square root right so you going to square
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root both Sid right so we going to
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square both side why do we stick the
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square root to get rid of the square to
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get rid of the Square thank you see
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that's how you think to get rid of the
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square because I'm trying to solve for x
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right but when I do that now what I have
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is this I'm going to have x + 3 = what
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plus or minus what six because I took
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the square root this is going to be plus
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orus
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6 because you can go both ways now I
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have to solve two equations right I'm
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going to have x + 3 is equal to 6 or
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what x + 3 = to -6 all right and then I
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can solve for
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x by doing what this is a simple
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equation subtract three here
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right we have x = 3
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right no it will be too loud for
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you right so we're going to have x 3 or
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X9 right so now what we're going to do
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is I have like two guided practice for
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you guys so I have now up here have X2
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right - 12x + 36 = 25 so what do I do
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shade what what do I do first oh I don't
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know I'm just taking notes how do we do
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this one uh put it into the square root
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okay so what do we do here put it into
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the square root okay so how would you do
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that you got to find two numbers that
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M right and add up to 12 what are they
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six and six and six thank you co so now
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we have what x - 6 s =
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25 right it goes 6 and 6 you notice that
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that's going to happen most of the time
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right six and six so we have EXT now
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what do I do here wait so do you have to
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find a perfect you want yeah yeah square
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root it right you take the square root
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of it so now we get what x - 6 Plus or -
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5 plus or - 5 and now we split the this
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two and we solve each one separately
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right so now we have x - 6 = 5 or x - 6
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= -5 now we have X = -11 and then here x
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= uh POS 1 would it be posit 11 CU
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you're adding six to the side no oh yeah
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yeah POS 11 right and then here postive
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1
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right that's easy that's pretty easy
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right now what about this one here how
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would you solve it we have X2 - 16x + 64
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=
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49 so how do you set that
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up x s x - 8 s = 49 49
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sare so square rooted right so we get x
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- 8
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= plus or- 7 plus or - 7 so we just
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going we can split them right so if you
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want to do go through that so and then
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the rest is just is and then X is
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negative 1 actually no POS 15 and then
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here x = 1 all right so you see this is
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how you solve this type of problem right
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so this this is called uh rational Roots
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so thus far this is easy because it's
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just straight up this is a perfect
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square this is a perfect uh uh perfect
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what do you call that um trinomial
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perfect identity so we can do that here
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we can not going to have to do that here
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as well right now this is different so I
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have X2 - 10x + 25 = 27 so what do I do
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here yeah you can use five and five five
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and five so that what
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equals to 27 and then what and then you
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do but is a perfect square it's not
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perfect square right now we have to
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simplify it's 9 and three right so this
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is still plus or minus 27 but now we
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have to work with this guy here because
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yes
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sir do you have like a marker that's
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like working yeah the red one yeah right
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so now the thing is yeah would be 6 no
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it's not six 27 is what 9 * 3 right so
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be 3un 3 so we can break this down right
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so we going to get x - 5 = > of 27 right
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or x - 5 = > 27 now what we can do here
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is we can break this down right 27 isun
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9 * 3 same thing here
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right 9 * 3 and now we can break down of
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9 is what 3 so we get x = X - 5 = 3 <
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tk3 right and here we got x - 5 = 3 <
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tk3 and then how do I solve for x from
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here add five add five right add five
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add five so that that gives me x =
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5+ 3un 3 and that gives me x = 5 minus
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3un 3 so what you have here is you call
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this rational irrational Roots right
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because these are not perfect squares
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all right these are no perfect squares
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so now I want you to try we're going to
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try two
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more and I think what I'm going to do is
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I'm going to save the actually maybe
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we'll see we'll
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see so do
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uh try this x² + 8
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x + 16 is = to 20 I want you to try that
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I also want you to try
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um
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X2 - 6 x + 9 equal 32 all right go ahead
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and try these
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two see what you get have your marker
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[Music]
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stilling yeah yeah I know that's fine
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just going to do this one and then shout
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it after that and then I'll give you
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guys some more to in the book cuz I feel
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like I want to stop here for today
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tomorrow we're going to actually talk
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about completing the square because
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that's different thing and I don't want
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to um I don't want to rush it
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is not perfect square so what we still
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have to simplif right excuse me oh no 16
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is not perfect
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either it's not six it's not you're not
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adding the numbers the closest perfect
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what four and what no 16 four four and
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five 4 * 5 is 20 right but what did you
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get for the left side 4 x + 4 s x - 4
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this is plus yeah right you got x + 4² =
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20 right which gives you x + 4 = 20 sare
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root or x +
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4 of 20 now we know that 20 is 4 * 5 so
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if you simplify that that gives you what
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x
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= 4 which is 2un of 5 or x + 4
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= No
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2 when you add our subtract no you can't
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add the four to theare root right we
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just leave it as it's going to be -4 -
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2un 5 and
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then4 + 2 5 you get it you don't you
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can't add because it's a square root
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number you can't add it to that you just
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to leave it as a radical right so X will
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be equal to -4 + 2un 5 4 x = -4 - 2un 5
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right you guys can go ahead and dle with
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this as well so it give you the same
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thing okay they give you something
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similar so