now what do we do just sqaure them?
#help-23
solid thorn
solid thorn
Yes
solid thorn
Wait can you explain where did this ratio come from?
neat yew
Similar triangles
solid thorn
Similar triangles
Which ones?
neat yew
AKL ABC
solid thorn
But where does AD/AE come from
neat yew
Heights from the same angle
solid thorn
Heights from the same angle
Alright
solid thorn
Heights from the same angle
Can you help out in 2 as well?
neat yew
Triangles are similar with ratio x/y
solid thorn
Triangles are similar with ratio x/y
AKL and ABC?
neat yew
AKD KBN
solid thorn
So ratio of areas is h^2 /9
Thank you so much, you helped me out a TON
Have a good day/night
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coral nimbus
i'm trying to figure out the equation(s) to get the lower and upper bound of a confidence interval around the mean at time t for a mostly-stochastic process. it is geometric brownian motion but with one slight difference in that there is a constant rate of change as well that is not subject to the randomness of the walk. how can that be integrated into the probability?
safe radishBOT
i'm trying to figure out the equation(s) to get the lower and upper bound of a c...
coral nimbus
if it is negative, it will have a dampening effect on the volatility, if it is positive, it will have an amplifying effect on the volatility.
without factoring in this constant rate of change, it is simple. but the thing i can't figure out is how it would effect the mean as well as the confidence interval around the mean once factoring in this constant rate of change.
here is a desmos to illustrate the problem: https://www.desmos.com/calculator/jrgujhwgdi
the red solid line is the mean (without the constant rate of change (M) factored in) and the red dotted lines are the confidence interval around the mean (again, without M factored in).
the blue solid line is my attempt at getting the mean with M factored into the equation. but i also need the blue dotted lines for the confidence interval around the blue mean.
finding this very difficult, as brownian motion is always described and talked about with at most 3 variables (drift, volatility, and a zscore), never something else like a constant fixed change per t.
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twin owl
Can someone help me with this
covert yoke
@twin owl I have no idea what the frogc Markov chain is. Is it something specific to this problem?
twin owl
Let me check
twin owl
<@704739879573127288> I have no idea what the frogc Markov chain is. Is it somet...
I think this
covert yoke
Makes sense
@twin owl do you know how to interpret a Markov chain transition table as a matrix?
covert yoke
Ok, you just say "it's a matrix now" and for any state vector v, then next state vector is Av
Then A^2 v and so on
The state vector is a column vector with the states in the same order as the state transition matrix
twin owl
okay i see
covert yoke
So for instance, the vector for (a) is (0, 0, 1, 0, 0, 0)
twin owl
how?
covert yoke
Because p_3 = 1
And all the rest are zero
(note that it's a column vector, even though I wrote it on one line)
twin owl
what is p3?
covert yoke
Can someone help me with this
twin owl
i only have to do C and D
covert yoke
Ok, so what would the column vector for (c) be?
twin owl
one sec let me just write the matrix down
safe radishBOT
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twin owl
Ok, so what would the column vector for (c) be?
im not sure
covert yoke
The column vector is (p1, p2, ..., p6)
twin owl
so would p2 correspond to the the second column in state in current period?
i dont understand what the question is asking
covert yoke
@twin owl tell me what you know about Markov chains. Do you understand what it is supposed to represent?
twin owl
yes
its bascially lays out the probablity of everything
ik how to make one based on a problem and how to make a tree diagram based on it
covert yoke
@twin owl it doesn't lay out the probability of everything
It is only the probabilities of state transition, specifically for memoryless random processes
If you feed it a vector of input probabilities then it feeds you back and output vector if probabilities
twin owl
okay
so for this problem do i replace the column p2 with the given values in the question
covert yoke
No
The Markov chain table is a matrix, we can call it A
And your input vector of probabilities is p = (p1, p2, ..., p6)
And to get the new probabilities you just multiply them Ap
@twin owl
twin owl
No
oh so matrix multiplication
i remember reading about this in the textbook
so how would i set this up?
twin owl
so the given matrix is A, i understand that but given the values of p2 do i make the same matrix and replace p2 with those vlaues and do matrix multiplicaiton?
covert yoke
So p1 is the probability that the system is in state 1. p2 is the probability that the system is in state 2, and so on
twin owl
yes
covert yoke
So (p1, p2, ..., p6) is the state vector that encodes everything known about the state
For (c), you have p2 = .25, p3 = .25, and p4 = .5
These add up to 1, so the probability of all other states must be 0
So your state vector is (0, .25, .25, .5, 0, 0)
twin owl
oh wait
i see
so you basically made a vector with the given values of p2
is that it?
twin owl
whats the correct terminology?
covert yoke
And I initially wrote it off as a typo
It's not that I'm correcting terminology, I'm trying to understand if you are misunderstanding a concept
p2 only has one value
covert yoke
It's the value of the probability that we are currently in state 2
covert yoke
Ok, just checking
twin owl
So your state vector is (0, .25, .25, .5, 0, 0)
i understadn this now but dont we have to multiply to get the new probabilities
twin owl
do p2, p3 and p4 (in C) correspond to p2, p3, and p4 in the columns or rows btw?
covert yoke
[
\begin{pmatrix}
.5 & .25 & 0 & 0 & 0 & 0 \
.5 & .5 & .25 & 0 & 0 & 0 \
0 & .25 & .5 & .25 & 0 & 0 \
0 & 0 & .25 & .5 & .25 & 0 \
0 & 0 & 0 & .25 & .5 & .5 \
0 & 0 & 0 & 0 & .25 & .5
\end{pmatrix} \begin{pmatrix} 0 \ .25 \ .25 \ .5 \ 0 \ 0 \end{pmatrix}
]
flat frigateBOT
OmnipotentEntity
covert yoke
@twin owl ^
twin owl
ohhhh
i get it so now we just perform the multiplication and will have the distrubution for the next period?
covert yoke
Yes
twin owl
ok j one sec while i do this
covert yoke
You miscopied the 4th row of the matrix
weak slate
can yall help me w geometery
covert yoke
As an aside, you can always check your multiplication for sanity to ensure that there isn't a mistake made by simply adding up your state vector. It should always sum to 1.
@weak slate ! occupied
! occupied
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Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
twin owl
covert yoke
@twin owl should always add to one. These clearly don't, because there is a 5 in the thousandths.
twin owl
im assuming d would be the same process
covert yoke
im assuming d would be the same process
Yes same process.
I have (1/16, 3/16, 5/16, 5/16, 1/8) which sum to 16/16 = 1.
(note your last answer was correct, you merely copied down the matrix incorrectly, I do not know how you arrived at the correct answer though. Maybe you made two errors that happened to cancel out, but only one was memorialized in text)
twin owl
covert yoke
Looks good to me
twin owl
tysm man
i think i understand this concept now
do you mind if i friend u and dm u if i have any other questions? i will use this server but you really helped clear things up
covert yoke
I don't mind, but I cannot promise I'll always be available, and you will likely find it much faster to get help via the server regardless.
twin owl
okay cool im going to work through part d now, would you mind waiting 5 mins so i can show u ? if not its fine i can do it now
covert yoke
I'm actually in bed about to sleep. So I might not be up in a few minutes. However, if I'm not there are online matrix multiplication tools. You can check your work with them!
twin owl
I'm actually in bed about to sleep. So I might not be up in a few minutes. Howev...
oh okay, ty for your help!
covert yoke
i dont think this is right
It actually looks correct to me
Some Markov chains have stable states
twin owl
really? i thought the values for the next period would be different but they can be the same?
covert yoke
This is one of them
twin owl
ok cool, thanks a lot man!
covert yoke
You're very welcome!
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sour valve
This is linear algebra: how can I check if my solution is right?
I know 2 solutions of Ax=(1,4,5) which are (1,3,-1) and (0,1,2) -in fact they are horizontal solutions-, A is a R3x3 matrix, I have to find 4 more. My result is (2,5,-4) (3,7,-7) (4,9,-10) and (-2,-3,8); any clue how can I check if this is right? How this makes sense lol
safe radishBOT
This is linear algebra: how can I check if my solution is right?
I know 2 soluti...
sour valve
Hope this makes sense as well
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distant monolith
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distant monolith
How should I solve this problem?
hasty wagon
How should I solve this problem?
looks fun
I'm thinking is it possible to do
f(x)= 1-2/(2^(2x)+2)
quick crater
I'm thinking is it possible to do
f(x)= 1-2/(2^(2x)+2)
i triwd that but i couldnt get anything from thaat unfortunately
hasty wagon
awwww
quick crater
maybe you could get something
hasty wagon
hmmm
hasty wagon
since it's a mutliple choice question, I'd try to guess XD
like we can change to the form
1-2/(2^(2x)+2)
then, we have 2022 1's.
consider, x=0 and x=1
x=0, 2/(2^(2x)+2) = 2/3
x=1, 2/(2^(2x)+2) = 1/3
By intuitive guess, I'll take the middle number between 2/3 and 1/3 and multiply by 2022, so we get
the sum = 2022-1011
but of course, this is only a guess....
I'll try to do it on my phone to verify
yea, i think i can verify
group:
first and the last
2nd and the 2nd last
3rd and the 3rd last
each grouped term add up to 1
hasty wagon
Yeah it does
@distant monolith you wanna try it out yourself or shall i demonstrate?
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magic tusk
$d(t) = 3cos(\frac{\pi}{6}t - \frac{\pi}{3}) + 5$
safe radishBOT
$d(t) = 3cos(\frac{\pi}{6}t - \frac{\pi}{3}) + 5$
flat frigateBOT
pixel
waxen inlet
Like normal functions, f(x - a) moves f(x) a units to right, and f(x + a) moves it a units to left
In your case f(x) is cos(pi/6t)
magic tusk
Like normal functions, f(x - a) moves f(x) a units to right, and f(x + a) moves ...
ya i know that
but by how much
cause like u see how x-axis is not in radians
its not pi or whatever
how would i know how much pi/3 is?
waxen inlet
pi/3 is still a number though?
It's not gonna be nice round numbers but you can still write them
Wait nvm mb
magic tusk
wait
i got it
u have to factor out pi on 6
which gives u t - 2
and that makes it translate 2 to the right
waxen inlet
Yeah that's correct I made a mistake in my solution sorry
solar rampart
[warning] this question is harder than it looks, at least I think so myself
Given a denomination of 50000, how many combinations can you divide it into 50000, 20000, 10000, 5000, 2000, and 1000?
Dividing 50000 into 50000 counts as one way.
20000, 20000, and 10000; and 10000, 20000, 20000 is only counted once, so they're not different.
safe radishBOT
[warning] this question is harder than it looks, at least I think so myself
Giv...
@solar rampart Has your question been resolved?
mortal sandal
We can divide everything by 1000
Aside from that I think these problems need to be brute forced with a computer
There's a solution with generating functions: look at the coefficient of x^50 in (1+x+x²+...x⁵⁰)(1+x²+x⁴+...+x⁵⁰)(1+x⁵+x¹⁰+...+x⁵⁰)...(1+x⁵⁰)
solar rampart
We can divide everything by 1000
yeah
solar rampart
Aside from that I think these problems need to be brute forced with a computer
man 😭 feels like the prof is lying
"this question is easy"
"the solution has to be elegant"
and i bruteforced 20000 and got the correct answer but it was not elegant LMAO
solar rampart
There's a solution with generating functions: look at the coefficient of x^50 in...
OHHHH i see
mortal sandal
Power series of 1/[(1-x)(1-x^2)(1-x^5)(1-x^10)(1-x^20)(1-x^50)] but I still don't see how that makes things nice
solar rampart
is it with each different denomination? it's not really practical but it's cool tho
mortal sandal
Yeah
solar rampart
Power series of 1/[(1-x)(1-x^2)(1-x^5)(1-x^10)(1-x^20)(1-x^50)] but I still don'...
ahhhh
i see though, so there's not really a quick and simple way to just solve this right? just various bruteforcing methods
solar rampart
ohh?
mortal sandal
Store a table of the number of ways to make k out of 1s and 2s
For 1≤k≤50
Use that to make a table of the number of ways to make k out of 1s 2s and 5s
etc
Like a dynamic programming solution
solar rampart
ohhhh
mortal sandal
There's a solution with generating functions: look at the coefficient of x^50 in...
Basically this though
Where you multiply through and don't bother storing the terms of degree >50
solar rampart
i see
oh wait
OH i just got the concept of the multiplication yeaa
that's neat though
solar rampart
There's a solution with generating functions: look at the coefficient of x^50 in...
i think this should be enough for the solution, i don't think there's an ultra simple way to solve this, thank you so much for the help!
mortal sandal
No problem
solar rampart
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lean hinge
Heelppp
icy lance
oops
flat frigateBOT
icy lance
,rcw
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exotic cypress
i think they're doing division wholesomely
not variable division to get factors
so the reason for long division here, good sir, is to try and break down the polynomial into (ax+b)(cx+d) if possible
icy lance
aha
exotic cypress
to make calculations, turning points, other stuff easier
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@lean hinge Has your question been resolved?
lean hinge
Hi, im trying to do long devision
exotic cypress
yes
so with x you need to times it by x to get a term with x^2
so how would you go from 3x to -10x^2?
lean hinge
so with x you need to times it by x to get a term with x^2
Yes
brisk sail
i hope you improve your internet speed. Thanks.
exotic cypress
helpers help, he's too slow for me i am gonig rn
lean hinge
so how would you go from 3x to -10x^2?
Add 7x?
exotic cypress
almost that gets 3x to 9x^2
ok simpler, how get x to 10x^2
then / that answer by 3
im off now gl with understanding
lean hinge
im off now gl with understanding
Okay, thanks for the help
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lean hinge
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soft matrix
When is nCr odd and when is nCr even 
peak estuary
lucas theorem
soft matrix
lucas theorem
dam bro ruined the ques
I still won't look up theorem since it was given to us to try it out ourselves and discuss with others to find out sol
Was here hoping for a hint
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tranquil brook
I have tried solving this question and my answer was answer choice c yet the answer module has it written down that the correct answer is b
safe radishBOT
I have tried solving this question and my answer was answer choice c yet the ans...
stable flint
!show
safe radishBOT
Show your work, and if possible, explain where you are stuck.
tranquil brook
To my understanding the range should start at the value that will make the sum amount to greater than or equal to 0
I thought 2 would be included in the range
As 2-1 is 1 and that is greater than or equal to 0
tranquil brook
X greater than or equal to 0
honest perch
sqrt(x-1)?
honest perch
X greater than or equal to 0
you mean [0, infty)?
tranquil brook
Yes
honest perch
yeah it's like g(x) >= 0 not x >= 0
tranquil brook
sqrt(x-1)?
[1,+infinity)
honest perch
no
tranquil brook
yeah it's like g(x) >= 0 not x >= 0
Ohhh
honest perch
sqrt(1-1) = 0 so 0 is in the range
tranquil brook
Okay so it's sqrt(x-1) =-3/2
honest perch
?
tranquil brook
Don't I solve the equation?
honest perch
what equation
tranquil brook
From the question I sent lol
tranquil brook
Is range the one on y axis?
honest perch
yes
tranquil brook
look at what y values are in the graph
[0,+infinity)
tranquil brook
Does * mean multiplied?
honest perch
yes
tranquil brook
Do I solve it by saying 2*sqrt(x-1) is greater than or equal to 0?
honest perch
no
tranquil brook
What is the formula sorry
honest perch
there is no formula
what's the range of this function
R(f) = {f(x): x in D(f)} if you want
tranquil brook
(-infinity,+infinity)
tranquil brook
I can tell from a graph I just have difficulty when it's just the equation
honest perch
this the graph of sqrt(x-1)
can you graph 2 times this
you're multiplying the y coord by 2
tranquil brook
Would it still be the same?
honest perch
for every point
honest perch
Would it still be the same?
same graph? no
tranquil brook
Cuz 0*2 is 0
tranquil brook
same graph? no
No same range
tranquil brook
Okay I've grasped the general idea I think
honest perch
fixing 0
tranquil brook
In the question I sent did it start at 3 bc the equation had +3?
solar hazel
mayer-vietorUs
honest perch
mayer-vietorUs
hello
solar hazel
hello
tranquil brook
Will you still help me with other questions or would I need to open a new channel?
honest perch
neither
when you +3 you add 3 to the y coord of every point so the graph shifts up by 3
tranquil brook
Yes I understood that
honest perch
you can do whatever you want
tranquil brook
Okay
How can I know the range amd domain for ln functions
What i got from watching yt is that the mothergraph is that the domain would be open parentheses to the x value of the line on the vertical line that the graph is approaching
Then +infinity)
Like ln(x) the domain for it is (0,+infinity)
And for range it would be (-infinity,+infinity)
exotic bolt
So
For the domain
There’s two ways to think about it
First the parent graph
lnx
which this graph is just ln (-(x-2))
So it has a horizontal reflection ( aka reflection across the y axis), and a horizontal translations of -2 units
or you could just
2-x>0
sacred sentinel
yes
tranquil brook
For the range is it (-infinity, infinity)
And if it is will it always be that way
exotic bolt
ye for logs the range is always that
tranquil brook
Okay ty
I know I can solve this by individually devising each equation to check but that would take too much time is there any easier way
<@&286206848099549185>
lean otter
perhaps check to see if any of the polynomials have x=-1 where P(-1) = 0
maybe that might work?
im not 100% sure tho
stable flint
(next time make a new channel for new ques)
young nexus
zeros of x+1 has to be zeros of P(x) if P is divisible by x+1
lean otter
(next time make a new channel for new ques)
its still the same sheet
tranquil brook
(next time make a new channel for new ques)
It's my first time using the dc and I asked a helper they said do wtv u want lol
tranquil brook
its still the same sheet
Yes it's just very long
stable flint
its still the same sheet
!1q
safe radishBOT
its still the same sheet
It is suggested that you limit yourself to one question per help channel, opening a new one once your original question is answered and your original channel has been closed. This is to make your channel easier to follow for potential helpers and can bring attention to the fact that your question has changed.
tranquil brook
perhaps check to see if any of the polynomials have x=-1 where P(-1) = 0
I will check rn
It worked
Ty
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uncut star
Hello!
rustic goblet
x^2 -2x <= 0
you need to find the domain and range of this?
uncut star
Yes, that's right
rustic goblet
but that's an inequality 
uncut star
And I need to say which sets that x is allowed to be in and all that good stuff
rustic goblet
you can only find the domain and range for functions
uncut star
Okay, I see
rustic goblet
And I need to say which sets that x is allowed to be in and all that good stuff
sure
rustic goblet
can you post the original problem? 
rustic goblet
but what are they asking you for?
uncut star
"solve the given inequality, given the solution set as an interval or union of intervals"
that makes more sense, but I am still lost
uncut star
Yes
rustic goblet
have you solved quadratic inequalities before?
rustic goblet
sure thing
uncut star
Once I do that, I can see that x = 2 would fulfil inequality
if that's how I can word it
also x=0
Now I need to think about the intervals, right?
rustic goblet
Once I do that, I can see that x = 2 would fulfil inequality
indeed it would, but we want to find all values of x such that the inequality holds
rustic goblet
Now I need to think about the intervals, right?
indeed
a sign chart, if you're familiar with those, would be of excellent help here
uncut star
I don
I don't know what a sign chart is
Is that where I can see what these funny symbols mean?
rustic goblet
a sign chart is a tool we use to figure out what intervals a polynomial function is positive and negative on, with only knowledge of its roots
like this
uncut star
Okay
rustic goblet
I would highly recommend using one here
you don't even need the graph
just the factored form of the quadratic is enough
uncut star
Alright that looks vaguely familiar seeing as the prof used something similar
Can I ask about the example he used?
rustic goblet
sure
uncut star
The problem I am doing is from the book
Okay he had this inequality
|x+3| <= 10
For me I am still confused on how he did this because
It looked so easy
And I want to do it as easily
He wrote this
uncut star
|x+3| <= 10
This expressions is equivelant to -10 <= x + 3 <= 10
I don't understand how he gets to that
I think most people understood but I didn't
Something about || being the distance between two points, but I still do not understand
is it because we remove the absolute value?
that -10 appears?
Sorry, I am just totally confused
uncut star
This expressions is equivelant to -10 <= x + 3 <= 10
He followed it up with this expression being equivalent to -10 - 3 <= x + 3 - 3 <= 10 - 3
That made more sense to me intuitively
so I am with him on that step
rustic goblet
|x + 3| <= 10 means "the distance between x and -3 is less than or equal to 10"
a picture may help
rustic goblet
okay, I'm stealing an example from the internet 
uncut star
okay, I'm stealing an example from the internet <:KEK:586240877358350341>
We love theft
uncut star
This expressions is equivelant to -10 <= x + 3 <= 10
Okay so I am not too bothered about the absolute value I think, more how he got to this expression instantly
I fear I may not even understand my own confusion though
He did write a line to try to explain to us
how you can visualise it
but he did that after arriving to that point
and I also don't understand the line to be totally honest
rustic goblet
hm, I feel like I don't actually know how to explain this in a way that'll not be confusing
@plucky elk if you're still here, do you know?
I only now realize that I lack the words to describe this idea properly 
uncut star
Try the confusing way if you want
Something that is also a "doubt" is that I don't understand how this would be applicable to situations where there is no absolute value, like the problem in my book
x^2 - 2x <= 0
You are just supposed to reason your way to the solution intervals?
rustic goblet
x^2 - 2x <= 0
this inequality might actually be easier for me to explain the reasoning
absolute values make things a little weird
uncut star
Yeah do that instead
I am curious 'bout either
And also thank you for taking the time
I am pretty dense
rustic goblet
you factored this to x(x - 2) <= 0 earlier, right?
rustic goblet
the way to do this is to mark the locations where the quadratic equals 0
rustic goblet
and then consider the intervals created by using those roots
uncut star
There is a 0 and 2
rustic goblet
yes
rustic goblet
I'm thinking of this rn
uncut star
I am trying to think about the <=
symbol
yes that is what I have
I am trying to interpret the <=
rustic goblet
alright
uncut star
Idk why but my mind is drawing a blank rn but give me a second
We can't have any negative numbers
rustic goblet
let's do the intervals one by one
uncut star
Wait
rustic goblet
okay
uncut star
No I am sorry go on, my brain is just trying to pull a prank on me
uncut star
Yep
uncut star
Positive and then there is debate if 0 is negative or positive
rustic goblet
oh, no no
uncut star
oh ok ok
rustic goblet
the 0 and the 2 I've marked are just the roots of the quadratic
uncut star
damn ok
rustic goblet
they'll serve as markers for our intervals
if x is less than 0, then x is surely negative, right?
rustic goblet
okay boom
uncut star
Lovely I love that
rustic goblet
we're going to mark that on this interval
now, what about (x - 2)
is (x - 2) negative in this interval?
uncut star
I am still trying to decide why it is positive: Let me think for a moment
rustic goblet
rustic goblet
sure
rustic goblet
we aren't done this interval yet!
uncut star
WAIT
rustic goblet
the interval is (-infinity, 0)
uncut star
yes
uncut star
you called that an interval
uncut star
now, what about (x - 2)
but then you called this an interval as well
rustic goblet
I did not call (x - 2) an interval
rustic goblet
I asked if it was negative on the interval (-infinity, 0)
rustic goblet
our ultimate goal is to find out if x(x - 2) is negative or positive on (-infinity, 0)
we're breaking this up into 2 smaller pieces
uncut star
I feel like it is a big problem that I don't understand why that is our ultimate goal
rustic goblet
which are to find if x and (x - 2) are positive and negative on the interval
rustic goblet
I feel like it is a big problem that I don't understand why that is our ultimate...
yeah, this problem requires you to break it down a lot
that's math at large too
rustic goblet
which are to find if x and (x - 2) are positive and negative on the interval
okay, so
uncut star
Right I am not sure how to reason around (x-2)
rustic goblet
if x is in the interval (-infinity, 0), then x < 0
rustic goblet
yup!
rustic goblet
if we subtract 2 from any negative number, we're going to get another negative number
uncut star
still don't understand why we are doing this, but you said we are breaking the problem down
rustic goblet
this means that (x - 2) is negative on (-inf, 0)
rustic goblet
still don't understand why we are doing this, but you said we are breaking the p...
it'll be clearer as we go
uncut star
if we subtract 2 from any negative number, we're going to get another negative n...
that makes sense to me
rustic goblet
just trust me on this :p
rustic goblet
okay, nice
now, let's figure out if x(x - 2) is positive or negative on (-inf, 0)
both x and (x - 2) are negative on this interval
uncut star
That would be positive
rustic goblet
and x(x - 2) is their product
rustic goblet
That would be positive
YES
uncut star
negative * negative = positive
uncut star
hmm okay
uncut star
can you explain what is going on here
rustic goblet
so x(x - 2) > 0 on (-inf, 0), so I mark that interval with a +
uncut star
because I was with you up until we put a 0 and 2 on the line
then I was already lost
No wait
x is <0 there
rustic goblet
the reason we choose 0 and 2 is because they are the roots of the quadratic
uncut star
and then x-2 is also <0
and x(x-2) is greater than 0
okay yeah no that does make sense on that part of the number line
rustic goblet
and only by passing through a root can a function change from positive to negative or vice versa
that's why we mark the roots
uncut star
brother
uncut star
and only by passing through a root can a function change from positive to negati...
that's crazy
uncut star
yeah that makes sense but it is still crazy to me
but wait
are we supposed to have solved it now?
is that the solution somehow?
because then I am still not quite there
rustic goblet
this one is next
rustic goblet
(0, 2)
uncut star
brother
so we just ignroe the roots
no we dont
we are only interested
in what is between them
rustic goblet
so we just ignroe the roots
well, we're gonna bring em back later :p
it's easier if we ignore them rn, because 0 is neither positive or negative
uncut star
ok
uncut star
I will think now
So our ultimate goal here is to determine
for some reason
if x(x-2) is pos or neg
and
rustic goblet
mhm
rustic goblet
if x(x-2) is pos or neg
to do this, we have to determine if each factor is positive or negative on this interval
the two factors are x and (x - 2)
rustic goblet
good
uncut star
and (x-2) is negative
rustic goblet
yup!
uncut star
because 1.9999999 -2 is still negative
or whatever
and then x(x-2)
mmm
negative?
nah idk
i havent reasoned
uncut star
yes it is
rustic goblet
last one!
rustic goblet
indeed
uncut star
a number greater than 2 - 2 would still be positive
x is positive
so it is positive
uncut star
god I feel like such a dumbass, but this has been very helpful so far despite me not being able to understand the purpose of all this yet
uncut star
Only in the interval (0, 2) ???
rustic goblet
yup!
rustic goblet
when does x(x - 2) = 0?
uncut star
it equals 0 ... at the roots?
uncut star
[0, 2]?
uncut star
XD
rustic goblet
do you see why the sign chart is so useful?
uncut star
I see why it is useful and I will continue to use them from now on
rustic goblet
it lets you think of everything one by one
uncut star
However I still think I need to practice using them because that was a little bit much for me to comprehend on my own
rustic goblet
that's okay!
uncut star
Yeah that broke everything down into smaller pieces
rustic goblet
you only get used to them through practice
rustic goblet
rustic goblet
yes
rustic goblet
factor, find roots, mark them on a line, sign chart
uncut star
what do I do if I cannot factorise
rustic goblet
all polynomials can be factored into linear and quadratic factors
this is a theorem
rustic goblet
if it is a linear factor, then we are good
uncut star
so (x^2)(43+x) is a problem?
rustic goblet
but if it is a quadratic factor which we cannot factor into two linear factors, then the quadratic will always be positive
regardless of x
rustic goblet
so (x^2)(43+x) is a problem?
well, you can write that as x*x*(x + 43)
so it is not a problem
rustic goblet
but you can just keep it as x^2 too
uncut star
what would be an example
rustic goblet
(x - 2)(x^2 + 1) is different
uncut star
ah
uncut star
so
the problem is almost easier
because we dont need to use a sign chart
for that part
of the interval?
or
something
rustic goblet
well, I would recommend using one anyways
because they're just good for bookkeeping
rustic goblet
no, sign chart them all
uncut star
okay uhmm
rustic goblet
the only thing is that you get x^2 + 1 > 0 for free every single time
you don't even need to think about it
rustic goblet
factor -> find roots -> mark roots on a number line -> sign chart
rustic goblet
(x - 2)(x^2 + 1) is different
let me do this one btw
uncut star
yeah, would you mind if we went through that one?
I just want to understand what you mean by (x^2 + 1) > 0 being given
rustic goblet
uncut star
(x^2 - 1) > 0
rustic goblet
there's only one root, and it partitions the number line into 2 intervals
uncut star
this is not given
uncut star
it can be factored but isnt given that it is greater than 0
uncut star
ok just want to make sure i understand
uncut star
yeah that makes sense because of the properties of multiplying negative numbers with one another
i also want to say that I only have to deal with real numbers at the moment
rustic goblet
I know
uncut star
it will be like that for quite some time as I understand things
rustic goblet
that's fine 
uncut star
and frankly I am not looking forward to the complex ones
rustic goblet
you'll get the hang of this soon
rustic goblet
and frankly I am not looking forward to the complex ones
in some sense, the complex numbers are easier to work with
everything is nicer
uncut star
I mean if that is true then I want to go to math land
rustic goblet
so I hope I stay in engineer land
engineers (especially electrical engineers) use complex numbers all the time
they are super useful
rustic goblet
sure thing
uncut star
and I will post results here
safe radishBOT
@uncut star Has your question been resolved?
uncut star
This is not the solution but I am a little bit overwhelmed by how I need to reason
to arrive at the solution
and I also worry I have done something wrong
uncut star
I look forwards to seeing what you can do <:happy:908558103992225822>
If you are still there I'd appreciate your help once more!
also sry for ping
uncut star
it is silyl but how do I arrive at the solution
I am supposed to reason here
like
"Where is this inequality greater than 0"
I have my sign chart
So I can see where it is greater than 0
That's lovely
But
rustic goblet
well, we need to find where x(x + 2)(x - 2) > 0
uncut star
Exactly
rustic goblet
take the left most interval
uncut star
And I think I have done that with the sign chart
uncut star
x < 0, x + 2 < 0, x - 2 < 0
This is referring to the right most part interval of the sign chart?
no wait
rustic goblet
uhh, the left most one
rustic goblet
you are not stupid btw
uncut star
so x(x + 2)(x - 2) is ?
less than 0?
no
equal to 0?
no wait
dont say
dont say
rustic goblet
I'm just looking for a word
uncut star
it is less than 0
rustic goblet
not a number
uncut star
negative
uncut star
i get it now i am just being silly
rustic goblet
neg * neg * neg = neg
rustic goblet
so mark the left most interval with a -
uncut star
right, i totally forgot about this
rustic goblet
second is negative too
it is?
uncut star
i will send u photo soon shouldnt be too long
rustic goblet
sure
uncut star
i will think twice about that one XD
uncut star
nah it is pos
i read it wrong
but i do see why it is pos
actually i wont send photo just say them
oh
rustic goblet
looks good to me
uncut star
okay and i still have trouble with the reasoning aspect
NO
WAIT
UNION
UNIOOON
WAIT
rustic goblet
‼️
rustic goblet
take your time
uncut star
if it said =>
i would check roots now
but it says >
so no need to check roots
include roots*
am I thinking correct about this?
anyway for the union
rustic goblet
(-2, 0) union (2, +infinity)
rustic goblet
you did it!
uncut star
that's awesme
rustic goblet
be proud! 
uncut star
let me jus tcheck what the book says
bro you should be proud as well you are really something!
thank you for helping me
rustic goblet
I try my best
uncut star
I have one more question
rustic goblet
sure
uncut star
this is
fast ( i think)
this is illegal right? just intuitively? [-infinity, +infinity]?
i have to say (-infinity, +infinity)?
rustic goblet
yes
uncut star
i cant "Include" infinity
rustic goblet
+infinity and -infinity are not real numbers
so we can't include them as being elements of an interval :p
rustic goblet
mhm
rustic goblet
now you do
uncut star
"factor, find roots, mark on line, sign chart"
idk if you came up with that yourself but that is a really good way to teach someone this
if they can just nod along for the part they dont understand
rustic goblet
I didn't come up with the method
uncut star
"factor, find roots, mark on line, sign chart, check where the criteria of the inequality is met"?
idk how you would word the last part so it stas concise
rustic goblet
I think that's fine
uncut star
anyway I am probably going to go home now and eat
rustic goblet
well deserved
uncut star
again really thank you, that felt really good
uncut star
i am a little late only started putting in effort about 10 monts ago
i hope there is still time for me to salvage my education and steer my life a more positive way
people like you increase the likelyhood of my success!
rustic goblet
i am a little late only started putting in effort about 10 monts ago
better late than never
you've got this!
you can do it!
safe radishBOT
rustic goblet
safe radishBOT
Available help channel!
quaint aspen
safe radishBOT
@quaint aspen Has your question been resolved?
dense wadi
safe radishBOT
wraith locust
x6 = 1/3
substitute to x3
so
x3 = 1-2x4-3x6
= 1-2x4-3(1/3), 3(1/3)=1
= 1-2x4-1
x3 = -2x4
solar hazel
wraith locust
dense wadi
x6 = 1/3
substitute to x3
so
x3 = 1-2x4-3x6
= 1-2x4-3(1/3), 3(1/3)=1
...
Okay so I multiply x6 = 1/3 3 times and then substitute right
dense wadi
<a:where_are_the_egirls_at:744880894497521735>
Hii Layla
solar hazel
hi :3