@eternal carbon can we continue doing this i had class but im back
#help-23
quartz wasp
eternal carbon
i think try reread my messages and take some time attempt on ur own
there’s not many ways to potentially use pythagorean here
you can show me ur progress later and i can give feedback
safe radishBOT
@quartz wasp Has your question been resolved?
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twin brook
Can someone help me with this again my teacher doesn’t want me to solve it by graph
safe radishBOT
Can someone help me with this again my teacher doesn’t want me to solve it by gr...
twin brook
Find all m that make the equation mx^2 + 2cosx = 2 has two different roots in [0,Pi/2].
How can we solve for m without graph
lean otter
do u know the discriminant
twin brook
Yeah
lean otter
yeah u gotta use it
faint seal
do u know the discriminant
this isn't a quadratic though..
lean otter
yeah i just noticed
twin brook
this isn't a quadratic though..
Exactly
faint seal
what values does cos(x) take between 0 and pi/2?
twin brook
I don’t understand the question
faint seal
I don't see an easy way to solve this without sketching graphs tbh
twin brook
Yeah it’s obvious with a graph
faint seal
I don’t understand the question
the question asks you to find the range of values of m such that the given equation has atleast 2 distinct roots
twin brook
Maybe it’s the only way?
twin brook
the question asks you to find the range of values of m such that the given equat...
Yup
faint seal
meaning, the graph of this function passes through the x axis twice in the given interval
twin brook
I get that
lean otter
Find all m that make the equation mx^2 + 2cosx = 2 has two different roots in [0...
also why say its a quadratic equation here btw
was this given to u in the question
is there something being misinterpreted
safe radishBOT
@twin brook Has your question been resolved?
lean otter
how to solve this
safe radishBOT
how to solve this
rose plume
which method you using?
lean otter
which method you using?
transformations
rose plume
what if you set sqrt(ax+b)-2 equal to 0? after all you need a 0/0
lean otter
what if you set sqrt(ax+b)-2 equal to 0? after all you need a 0/0
u mean ax+b-4 ?
rose plume
no
maybe you can rewrite the limit if you make both numerator and denomitar equal to 0
lean otter
maybe you can rewrite the limit if you make both numerator and denomitar equal t...
can we just Equalize ax+b-4 to x+3 ?
if the limit exist , ax+b-4 should equal x+3
rose plume
I think that also works
lean otter
ax+b=x+3
how can we solve this now
rose plume
can you show in which step are you?
lone arch
if the limit exist , ax+b-4 should equal x+3
Why?
I'd suggest to to multiply numerator and denominator by sqrt(ax + b) + 2
lean otter
can you show in which step are you?
lean otter
Ideally we want to get the x + 3 away..
yeah
only way is by assume ax+b-4 = x+3
to cancel them out
lone arch
only way is by assume ax+b-4 = x+3
Yep, but after your logic, a = 1 and b = 7. Then we get 1/(sqrt(1*(-3) + 7) + 2) = 1/4 as the limit
So that was already a good choice, but we need to tweak it to get -1/4
Notice that regardless of the choice of a, b, the denominator will always be positive after it cancels out with x + 3
We need the negative term to come from the numerator
So we need ax + b - 4 = ...
lone arch
I would start like this
Yep, contuining what we started is also possible though
lone arch
only way is by assume ax+b-4 = x+3
This was already good but we need to get a - in the numerator
So just make it = -(x + 3)
lean otter
Ive checked the question, and the answer is 1/4, not -1/4. There was a mistake
thank u
lone arch
Ive checked the question, and the answer is 1/4, not -1/4. There was a mistake
Ok, if it's 1/4 then what you said
ax + b - 4 has to be x - 3 to cancel it out
lean otter
yeah
rose plume
are you sure ?
lone arch
and then we just need to check if it's actually 1/4 what we get with that a and b
lean otter
are you sure ?
yeah
rose plume
ok
glossy obsidian
Hello does somebody know how to resolve this question : Let h be a function of domain R such that h'(x) = cos x - e^(sin(x)^2) and h(0) = 0. Using Lagrange's Theorem, show that
h(x) ≤ e * x, for all x ∈ R+
safe radishBOT
Hello does somebody know how to resolve this question : Let h be a function of d...
finite yacht
Hello does somebody know how to resolve this question : Let h be a function of d...
There are many theorems which are called Lagrange's theorem
Could you state that one?
glossy obsidian
It's the one that states that if we have a continuous function in the interval [a,b] and differentiable in [a,b] (the brackets should be reversed but idk how to type that) so it exists a c in that interval such as f'(c)= (f(b) - f(a)) / (b - a)
finite yacht
It's the one that states that if we have a continuous function in the interval ...
Maybe try to prove that e^x - h(x) ≥ 0 because (e^x - h'(x))(x-0) ≥ 0
That might work
glossy obsidian
Maybe try to prove that e^x - h(x) ≥ 0 because (e^x - h'(x))(x-0) ≥ 0
What lead u to do that
finite yacht
f'(c)= (f(b) - f(a)) / (b - a) so f(b) - f(a) = f'(c) (b-a). Then take f(x) = e^x - h(x)
However I forgot the -f(a) so I think that won't work
safe radishBOT
@glossy obsidian Has your question been resolved?
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dusky harbor
I have a question where they ask me to find the inverse matrix of a given matrix, where x is not equal to 0.
I almost got the correct answer, but I got a negative 15 rather than a positive...
safe radishBOT
I have a question where they ask me to find the inverse matrix of a given matrix...
dusky harbor
I'll show my working out, and perhaps you could troubleshoot what I did wrong.
I multiplied the top row by 5, the middle row by 3, and the bottom row by 1/x...
(-1/5)th of R1 was added onto R2...
I swapped R2 and R3 around, then multiplied the bottom row by -5
light shoal
shouldn't you be adding (1/5) of R1 to R2, not subtracting
otherwise you're not gonna get rid of the -1 in the first column of row 2
also, side note that i'm not sure is an issue here, be careful about dividing by x if you're not sure it's nonzero
dusky harbor
Hmm unsure? the -(1/5) of Row 1 would be -1 0 -1 | -1 0 0...
light shoal
(of course the matrix won't be invertible if x is zero, so maybe the assumption is that it's nonzero)
dusky harbor
Then adding.. wait
light shoal
Hmm unsure? the -(1/5) of Row 1 would be -1 0 -1 | -1 0 0...
sure, but then if you add...
you get -1 -1 = -2
vivid veldt
Out of nothing how to know if a number is a prime number ?
safe radishBOT
Out of nothing how to know if a number is a prime number ?
vivid veldt
I'm trying to upgrade my reasonning skills but i can't figure it out how to know if 143 is prime for example
I could just try every prime number I know from 2 to sqrt(143) BUT Fuck man i'm lazy
median vigil
vivid veldt
If it doesnt have factor other than 1
How I should prove that this number doesn't have a factor ?
median vigil
not an easy problem, the fact that it's hard to check that is the basis for a lot of computer science
vivid veldt
https://en.m.wikipedia.org/wiki/Primality_test
My teacher want to do this exercice wihout a calculator
vivid veldt
not an easy problem, the fact that it's hard to check that is the basis for a lo...
You're right my bad I would just do a python script for that stuff but well... Seems like it's possible wihout computer science ?
median vigil
you can just apply the same algorithms by hand
vivid veldt
Because doing like if 22 405 is prime (wait it's not since it got 5)
vivid veldt
you can just apply the same algorithms by hand
what about like 445599 ?
Just add them to see if it's divisible by 9 or 3 ?
median vigil
well in that case it's fairly simple to see that the sum of digits would be (4+5) + (4+5) + 9 + 9 which is clearly a multiple of 9
vivid veldt
Well while it's not a bad idea... I still haven't figure it out how to know if 143 is prime
vivid veldt
well in that case it's fairly simple to see that the sum of digits would be (4+5...
Ooo not bad
I have though to decompose the number into a sum
but I don't know what to do with this sum ?
I feel like this maths chapiter will piss me off
Arithmitic isn't my strengh...
vivid veldt
I could just try every prime number I know from 2 to sqrt(143) BUT Fuck man i'm ...
about using sqrt wihout calculator well u have to learn every perfect square ?
and do a "approchement" for knowing when to stop doing the algorithm by hand ?
median vigil
special cases for digits:
- if it ends in an even digit it's even -> not prime (except 2 ofc)
- if it ends in a 5 it's divisible by 5
- if the digits sum to a multiple of 3 then it's divisible by 3
vivid veldt
special cases for digits:
- if it ends in an even digit it's even -> not prime (...
Hm basic roles set... With this I just have to prove these 3 wrong and then I can conclude that the number isn't prime ?
Oh or just one condition
median vigil
if it fails one of those tests it isn't prime bc it's divisible by one of the mentioned numbers. if it passes then you have to do more checking by hand
vivid veldt
Alr thanks u
I will try
I appreciate your help really
:D
alr all 3 condition are false
So I have to do it by hand
from 2 3 5 7 11 ?
median vigil
in the case of 3-digit numbers you can check divisibility by 2,3,5 with the digits tests then every multiple of 6 +/- 1 (5,7, 11,13, etc.) until you reach a number whose square is larger e.g. 12² = 144
vivid veldt
in the case of 3-digit numbers you can check divisibility by 2,3,5 with the digi...
wdym with every multiple of 6 ?
median vigil
the 6k ± 1 rule mentioned in the article
vivid veldt
isn't the general role is 1 < p < sqrt(n) if the number isn't prime ?
vivid veldt
the 6k ± 1 rule mentioned in the article
Oh I wonder if I'm allowed to use it but yeah
vivid veldt
in the case of 3-digit numbers you can check divisibility by 2,3,5 with the digi...
Alr thanks u !
I will keep this message safe with the wikipedia link (I should use it more often)
.close
marsh walrus
I have kind of a silly question. I'm working on showing that if $\emptyset \neq S_1 \subset S_2 \subset \mathbb R_e$ then $\sup S_1 \leq \sup S_2$. Clearly $\sup S_2 \geq x$ for all $x \in S_1$. But my professors proof jumps to $\sup S_1 \leq \sup S_2$ from here. Isn't this an assumption of what we want to show?
safe radishBOT
I have kind of a silly question. I'm working on showing that if $\emptyset \neq ...
flat frigateBOT
jan Niku
marsh walrus
I get that it just makes sense but im trying not to do proof by obviousness
maybe a contradiction is actually the way to go
clear blade
the least upper bound of S1 is <= any other upper bound of S1
by the definition of l.u.b
marsh walrus
okay
yea i guess
idk why its not sitting with me
but ill buy it
thanks
.close
safe radishBOT
safe radishBOT
Available help channel!
crimson sage
safe radishBOT
crimson sage
I haven't really got a clue.
dire fjord
hint: check if there are lines which match the equations that you want to solve
in other words match each if your 2 equations to 2 different lines in the figure if possible
crimson sage
so the top to blue and the bottom to pink right?
safe radishBOT
@crimson sage Has your question been resolved?
safe radishBOT
@crimson sage Has your question been resolved?
upbeat ridge
Hello
upbeat ridge
Why is the first and third incomparable?
fiery merlin
What does weaker mean here?
safe radishBOT
@upbeat ridge Has your question been resolved?
upbeat ridge
What does weaker mean here?
Having less satisfied environments
fiery merlin
Well, if you only have <=>, it's equal. Then, you can leave it there or strengthen or weaken it in some rows. If you do that solely in one direction, it becomes that. If you do that in both directions, it's incomparable.
It doesn't want you to have ambiguity, where some rows weaken it and some strengthen it.
You're only allowed to mix equal and stronger rows or equal and weaker rows. Otherwise it's ambiguous.
Your example has a weaker row and a stronger row, so it's ambiguous, so it's incomparable.
upbeat ridge
You're only allowed to mix equal and stronger rows or equal and weaker rows. Oth...
It looks like they mixed weaker and stronger rows?
fiery merlin
Yes, the first row has p <-> q stronger than p, I think.
And the third row has p <-> q weaker than p, I think.
So, it's not universally pulling it away from equal in one way.
Oh, sorry, I think I got them backwards.
Either way, the idea is that you're only allowed to pull in one direction away from equal, either weaker or stronger, and if you mix them, it's incomparable.
safe radishBOT
fiery merlin
No problem.
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Available help channel!
velvet wave
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velvet wave
ummm idk
fading root
well idk if my answer is right then but what I did was I found out the length OR with pythag and OT would be the same length cos radius of circle
so both the triangles enclosed by the radii have area of 4
and the area of the sector is just the angle times the radius
which was not a nice number
but when you subtract the area of the sector from the area of the square you get that
twilit nebula
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jolly shard
What’s the question?
lean thorn
hi can you elaborate on what you're stuck on?
twilit nebula
whats the slope of the line
daring crescent
!status
safe radishBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
ok, if you have no further questions use .close
twilit nebula
.close
granite cape
safe radishBOT
granite cape
Could someone help me out with this?
jolly shard
Is the y axis n?
jolly shard
Could someone help me out with this?
!status
safe radishBOT
Could someone help me out with this?
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
1
x-axis is the number of matches won and the y-axis is the number of players
<@&286206848099549185>
.close
fallen grotto
Show that the following is an example of a maximal ideal: In F(R), the ideal J = {f:f(0) = 0}.
safe radishBOT
Show that the following is an example of a maximal ideal: In F(R), the ideal J =...
fallen grotto
I have: suppose there exists an ideal K between J and A. Note that if $g \in K$ and $g(0) \neq 0$, then $h(x)=g(x)-g(0)$ is in J since $h(0) = g(0)-g(0) = 0$. Theefore, $h, g \in K$
flat frigateBOT
Fuzzy_Alpaca
fallen grotto
Not sure where to go from here. They suggest using the idea that if J is an ideal of A and J contains an invertible element a of A, then J=A
<@&286206848099549185>
safe radishBOT
@fallen grotto Has your question been resolved?
gloomy light
safe radishBOT
gloomy light
I know the GIC is worth 1232.791534
So I take away 4000 to see how much he would need
Which is 2767.208466
But
The answer doesn't match up to the answer key or seem reasonable
safe radishBOT
@gloomy light Has your question been resolved?
gloomy light
I give up goodnight
safe radishBOT
@gloomy light Has your question been resolved?
granite cape
Hey guys, could someone help me out with this?
safe radishBOT
Hey guys, could someone help me out with this?
granite cape
I'm not the best with probabilities, just more confident with algebra and calculus.
<@&286206848099549185>
pseudo raptor
you were quite early
15 min countdown from the time you posted your problem ends now
idle widget
This isn’t a probability problem@i don’t think
It’s just proportions
You have a histogram
And they need X amount of players in the 16-21+ categories so that X=0.5T where T is their total player count
I guess they already have a certain amount of players in that category so just take those out with a difference
granite cape
Wait so they need 50% of their players to win over 15 matches which corresponds to the 16-20 and 21+ category
What would be my next step?
do I include the players in either Team A - 2021 or Team A - 2022, or both?
<@&286206848099549185>
Just really don't understand this question
<@&286206848099549185>
daring crescent
I think it would just be 2022
granite cape
Oh ok, but how would I start?
daring crescent
start by adding up all the wins for Team A - 2022 and then add up the 16-20 + 21+ and set that as a fraction, (15+)/total then simply add for like the first 8 to both (15+ + 8)/(total + 8) as it increases both numbers, you should end with a fraction that can be reduced to 1/2
Let me know what you get
granite cape
Why do we add 8 to both sides?
granite cape
start by adding up all the wins for Team A - 2022 and then add up the 16-20 + 21...
Sorry, I don't really understand the reasoning behind this.
daring crescent
so you are looking for a 50% or 1/2
the top is the amount of people who meet the requirements
the bottom is the total people
if you add people to the top you undoubtedly have to add it to the bottom as it changes the total amount of people.
if you see where the fraction is right now, it can give better insight into what number you're looking for, for instance you can't have an odd number in the denominator as it won't simplify to 1/2 so you can cross off any (total) + that # = odd #
once you figure out what numbers work from the answer you can add them to the total number of people and then divide that by two and you will get the amount you should now have in the requirements
(total + new)/2 = required amount for 50%
daring crescent
start by adding up all the wins for Team A - 2022 and then add up the 16-20 + 21...
simplifying to this simplifies the amount of work to do
does this analogy make sense @granite cape
granite cape
Oh yeah, that makes sense
granite cape
(total + new)/2 = required amount for 50%
Although this part is a bit confusing
granite cape
(total + new)/2 = required amount for 50%
I understand understand why we add 8, but if we do this:
daring crescent
lets see if I can use an example
granite cape
it would be (82 + 8)/2 correct?
daring crescent
yes
granite cape
which is 45
daring crescent
and then that will tell you what the number (original required + new) should equal
if it does not equal that then 8 is not the correct choice
daring crescent
so like lets say you have 50 people and you add 6, thats 56 in total
50% of that 56 is 28, hence we divide by 2
if we started with 22 people who were in the requirements and add 6 more that equals 28
so 6 is the right amount to add to get 50%
but if you started with 20 people who were originally in the requirements, thats 26, which would in this case mean 6 is not the right amount
@granite cape does this make sense
granite cape
Yes that makes sense, thanks for explaining :)
shell whale
Im lost at the inductive step after you replace n with n+1.
safe radishBOT
Im lost at the inductive step after you replace n with n+1.
shell whale
Looks like the next step is to distribute the left side, but I dont know how to procede after that
I assume the 2^n is from the right side of the assumption? but how do we also get +2?
Also confused about how to get to the next step too
rigid inlet
they expand the left side to get 2n + 3, then regroup as (2n+1) + 2
since you assume 2n+1 < 2^n, it's a useful rewrite
shell whale
they expand the left side to get 2n + 3, then regroup as (2n+1) + 2
can you explain more of this expand? can i assume thats the same ass distributing
rigid inlet
yes
shell whale
ok then. now how do we get to <= 2^n +2?
rigid inlet
they replace 2n+1 with 2^n, using a <= instead of the usual = we tend to see in other substitutions
shell whale
not sure how to get that outcome
so if 2n was like 11n+4912, i could also sub that with 11^n
rigid inlet
not necessarily
part of the inductive step is to assume 2n+1 < 2^n, so you're allowed to make use of this in the proof
rigid inlet
2 <= 2^n
what they're ultimately showing is that since a <= b <= c, then a <= c
and it just so happens that a and c are the things you're especially interested in
shell whale
.close
safe radishBOT
safe radishBOT
Available help channel!
spare grotto
Can anybody help me solve this, these are the only formulas that we are allowed to use on the test.
safe radishBOT
Can anybody help me solve this, these are the only formulas that we are allowed ...
solar quartz
write 2 as log
spare grotto
Like 2log with the base 2?
spare grotto
Oh okay
lime dust
write log(a)
Use a instead of x, this is something already used in the equation and can lead into misinterpretation
solar quartz
right, sry
lime dust
Show your work
lime dust
How much is log_2(2)?
spare grotto
I don’t really know
lime dust
How can you work with logarithmic equations then?
spare grotto
I am pretty new to this but we are not allowed to use calculators
lime dust
How much is log_2(2)?
You don’t need calculator to know this
spare grotto
Is it 1?
spare grotto
Like this?
spare grotto
Why is it not log_2 2, don’t we want to get it to 1?
lime dust
I want you to write as log_2(a) where a is a natural number
How do you know log_2(2) is 1?
spare grotto
Because 2^1 is 2
lime dust
Si which number do you get when it’s not 1 but 2?
spare grotto
4
lime dust
What is a then
spare grotto
4?
lime dust
Yes
lime dust
Yes, now you can substract log_2(4) in both sides of the equation and after that, in the RHS use, from your table
In the square 3 line 2
You will see what you have to do
zenith terrace
hey guys i need help, im having trouble understanding factoring by grouping
lime dust
Right, parenthesis are the mistake
wanton bough
The final answer is actually correct, you just made some mistakes in the process
zenith terrace
The final answer is actually correct, you just made some mistakes in the process
yeah thats what i dontunderstand
what should i change
zenith terrace
so i'll change the second line into + 9 ( + 3 )?
lime dust
What does that mean
wanton bough
-(9x + 27) becomes -9x - 27 when you simplify the parentheses.
lime dust
-a+b = -(a-b)
wanton bough
That's right
zenith terrace
-(9x + 27) becomes -9x - 27 when you simplify the parentheses.
im kinda lost
lime dust
im kinda lost
Read what I said
wanton bough
-(a+b) = -1(a+b) = (-1 times a) + (-1 times b) = -a + -b = -a-b
zenith terrace
What is -1 + 2?
1
zenith terrace
What is -(1+2)?
-3
zenith terrace
What is -(1-2)
it's 1
lime dust
Good
So the thing is that when u use parenthesis
You apply the - sign to all parenthesis
That’s why when u have -9x+27
If you want to take the -1 as factor
You must take care of the signs inside
-1(9x-27)
Because that’s equal to -9x+27
zenith terrace
ohhh i see
lime dust
Which is what we had
zenith terrace
lime dust
Now you take out -1 as common factor in the right
zenith terrace
Because that’s equal to -9x+27
ahhhhh
wanton bough
-1 times what is -9x?
wanton bough
1x
It's 9x, the sign gets changed. Similarly, -1 times a number is 27, that is -27.
-1 is a common factor in both of these so you extract it out of the bracket which makes it -1(9x + -27) = -(9x-27)
zenith terrace
It's 9x, the sign gets changed. Similarly, -1 times a number is 27, that is -27....
okok i see it now
loud jolt
safe radishBOT
loud jolt
Please someone tell me where is BKDL square
hazy elbow
BKDL square?
loud jolt
Yes
hazy elbow
can you translate the original question and send it?
loud jolt
buoyant shadow
,rccw
flat frigateBOT
loud jolt
I can't seem to connect those dots to get quadrilateral or whatever it's called
Someone help
Nvm
.close
frigid coyote
help
frigid coyote
help me draw the figure, (I got the given and whats to be proved)
obsidian violet
late eagle
Need help with this one, I don’t know what formula to use for letters b-e, any help is appreciated
safe radishBOT
Need help with this one, I don’t know what formula to use for letters b-e, any h...
lean otter
Need help with this one, I don’t know what formula to use for letters b-e, any h...
is the little circle with the denoting that C is the centre of the circle?
late eagle
Yes
lean otter
Yes
well i think for RZS at least:
this fact is important by the prependicular bisector theorem
we know the radii are equal (by definition)
so the chord SR must also be equally divided
now, it being bissected also implies that the angle SCZ is equal to the angle ZCR
so, also, the arcs SZ and ZR are equal
does this nudge you at the right direction?
(also someone verify what i said in case my geometry is incorrect)
late eagle
so, also, the arcs SZ and ZR are equal
I see
So the answer IS 50 degrees, many thanks! For letter b atleast
lean otter
So the answer IS 50 degrees, many thanks! For letter b atleast
no it shouldnt be 50
you are combining both arcs
lean otter
yes
late eagle
Alrightt, thankss
lean otter
do u know how to do c
late eagle
No
lean otter
just use the fact that
The measure of an arc is the measure of its central angle
you know the measure of the arc
so the measure of the angle is...
late eagle
50?
lean otter
yes
late eagle
I mean there’s d and e left
late eagle
Octa <:xd:1087388248101355520>
How bout octo for octopus
lean otter
I mean there’s d and e left
so like
for d
think about
the triangle TSC
can you tell me what angles CTS and SCT are
late eagle
I think angle t is 90 since it makes a right angle
lean otter
okay so like
dont just say "angle t"
there is like
5 different things that could be describing
late eagle
Oh
lean otter
but yeah its angle 90
late eagle
Aightt
And for SCT it’s 50 since it’s the center
Do I have to add it up then minus them by liek 180?
lean otter
Do I have to add it up then minus them by liek 180?
yeppies
lean otter
sounds legit yeah
for e
its a bit fishy because idk how to prove AB = SR 
ah they must be equal
but yeah i think you can show AB = SR easily
late eagle
Hmm so.. it’s also 8 then?
lean otter
alright no worries have a nice one
late eagle
Uh wanna be friends btw
lean otter
Double check in case btw. There is probably something more rigorous to what i said pretty much. I don't think i worded all of that concisely
late eagle
Yea
lean otter
but idk if your school cares about that
late eagle
Double check in case btw. There is probably something more rigorous to what i sa...
I think for e there’s a formula for it, not just simply a 2TS, I’ll probs watch more yt vids about chords justtt to be sure
lean otter
im pretty sure it should be 2TS but i could be wrong, not like geometry is my greatest suit lol
late eagle
Fair enough, math as a whole isn’t my strongest suit
lean otter
bye
late eagle
.close
safe parrot
Factorise $p(x) = x^3 -2x-4$
safe radishBOT
Factorise $p(x) = x^3 -2x-4$
flat frigateBOT
Merineth
safe parrot
How in the world should i attempt this? There isn't a common factor anywhere between all three terms
faint seal
rational root theorem
ruby bolt
take factors 4
safe parrot
I'm not familiar with rational root theorem
obtuse jackal
then become familiar with it
safe parrot
Is it any of these?
obtuse jackal
well obviously not
safe parrot
Then i'm not supposed to use it
Is there another way to solve it that doesn't use rational root theorem?
obtuse jackal
**Merineth**
"I hereby notice that x=2 is a root"
thin bridge
second one is related
rational root theorem by itself gives you candidates for nice roots (will definitely be allowed)
combine that with factor theorem and division
humble helm
"I hereby notice that x=2 is a root"
i use horners method cuz I'm lazy to learn these divisions or etc
obtuse jackal
,w Horner's method
flat frigateBOT
obtuse jackal
meeting expectations I see
faint seal
A lazy person would absolutely not use Horner's method
ruby bolt
just take factors of 4 and u get 2 as a zero
(x-2) - one factor
then use synthetic division
faint seal
then use synthetic division
no need for that
safe parrot
Kind of hard to use the factor theorem when i don't understand it :(
safe parrot
"I hereby notice that x=2 is a root"
Did you just try random numbers to see which reuslted in = 0?
faint seal
Did you just try random numbers to see which reuslted in = 0?
not random numbers
obtuse jackal
A lazy person would absolutely not use Horner's method
honestly yeah
obtuse jackal
Did you just try random numbers to see which reuslted in = 0?
2 looked extra sus
Other contenders that I would have tried in order would have been -2, 4, -4, then ask myself questions
i.e. just have the rational root theorem built in
thin bridge
Did you just try random numbers to see which reuslted in = 0?
not random
obtuse jackal
well yes there's also {0, -1, 1} but these were obviously not going to work
thin bridge
rational root theorem is what guides you to reasonable numbers to try
humble helm
A lazy person would absolutely not use Horner's method
I'm about this, it's when the roots are good idk
safe parrot
Interesting, I’ve never encountered this before
thin bridge
someone found a way to save more ink than synthetic divison
safe parrot
Like how am i expected to read these formulas
Pn(z0)
It doesn't say what P, n or z is
I assume P is polynomial
faint seal
$P_n(z)$ is a polynomial of degree n, $Q_{n-1}(z)$ is a polynomial of degree n-1
flat frigateBOT
kheerii
faint seal
$z_0$ is some random complex number and $P_n(z_0)$ is the value of the polynomial $P_n$ at the point $z=z_0$
flat frigateBOT
kheerii
faint seal
basically this says that if a complex number is the root of a polynomial P(z), then (z-z0) is a factor of the given polynomial
safe parrot
I'm having a hard time grasping what that even means even after several times reading it
So it's saying that if a Polynomial P of grade n when given a z0 (z0?) results in 0, then it's has a root there?
faint seal
well yeah, that's what being a root means
but the theorem is actually saying that if a polynomial has a root at z0, then (z-z0) is a polynomial factor of the polynomial
ruby bolt
roots are for equations
faint seal
for example, say we have the polynomial p(x)=x^2-4x+3
ruby bolt
Correct terminology is zeroes
faint seal
for example, say we have the polynomial p(x)=x^2-4x+3
here, if you check, we will get p(1)=0
safe parrot
And what was z0 again?
faint seal
And what was z0 again?
z0 is an arbitrary complex number
faint seal
here, if you check, we will get p(1)=0
p(1)=0 means that p(x) can be written as (x-1)q(x), where q(x) is another polynomial
safe parrot
oh okay, but what what is the difference between z0 and z? Just that they are different arbitrary numbers?
faint seal
oh okay, but what what is the difference between z0 and z? Just that they are di...
z is a variable and z0 is a particular value
say we have a polynomial $P(z)=z^2-4z+3$
flat frigateBOT
kheerii
faint seal
let me change it to z to match what is written
here, P(z) is a function of z
meaning as we vary the value of z, the value of P(z) will also change
z is a variable here
if we say $z_0$ is some arbitrary constant, then $P(z_0)=z_0^2-4z_0+3$ is also a constant
flat frigateBOT
kheerii
faint seal
do you get what I am saying?
safe parrot
Somewhat, yes
faint seal
so, with this same polynomial can you tell me what $P(z_0)$ will be at $z_0=1$?
flat frigateBOT
kheerii
safe parrot
No i can't :( i can't really speak math so it's hard to translate into words so i can understand
I'm not sure why it has to be written so complicated
I can't even read it properly
sleek wolf
**kheerii**
follow what you're being asked to for now
safe parrot
p(1) = 0
sleek wolf
ye
per what is being said here,
z_0 is a(n unchanging) number, making P just a number as well
safe parrot
What?
sleek wolf
Like how am i expected to read these formulas
I mean, we're just clarifying what this is about
safe parrot
That wasn't very clarifying 
sleek wolf
all that was only about the difference between z_0 and z :p
sleek wolf
"n" is the degree of the polynomial
"z" is the polynomial's variable
"z_0" is one potential value of "z"; in this case, a value that makes P evaluate to 0
safe parrot
Okay so it's saying:
If the polynomial of grade n is given a variable and results in zero, then it's a root.
and what is the difference between z_0 and z?
just to reference two different numbers?
ruby bolt
z is a variable and z0 is a particular value
.
sleek wolf
Okay so it's saying:
If the polynomial of grade n is given a variable and result...
ye, but that's only the "P_n(z_0) = 0" part
thin bridge
use c,k or something else if the presence of z in z_0 is confusing you
ruby bolt
Yeah WELL i don't know what a particular value is
Certain value
sleek wolf
that's why we just assume it
thin bridge
some constant
sleek wolf
it's just a number z_0 that makes P equal to 0
safe parrot
okay so some variable - some constant
sleek wolf
in this context, there aren't multiple variables, don't worry
so (variable - some constant)
safe parrot
And this part??
Now we introduce Q?
variable z - some constant times Q with a degree lower than Pn
sleek wolf
Q is another polynomial
imagine P divided by (z - z_0)
it's guaranteed to give you another polynomial which we call Q
oh, clarification question: can you determine the degree of (x^4 - 2x^3 + 3x + 2)?
the answer is 4 indeed
if P has degree 4, Q will have degree 3
if P has degree n, Q will have degree n - 1
safe parrot
That makes sense
sleek wolf
that Q, multiplied by (z - z_0), will make P
safe parrot
So how am i supposed to use the factor theorem to solve my factorisement?
sleek wolf
- find any z_0 that makes P, your polynomial, evaluate to 0
- divide P by (z - z_0)
you'll make instant progress and have a factor
sleek wolf
I'm sure (x^3 - 2x - 4)/(x - 2) has a different answer
safe parrot
I'll try again, one moment
Having a hard time once again with polynomial divison :(
It this not how i start?
sleek wolf
oh no ;-;
safe parrot
How many times x fits into x^3
sleek wolf
x^3 / x = x^2
sleek wolf
and then you multiply x^2 by (x - 2), and subtract the result from x^3 - 2x - 4
safe parrot
i see what i did wrong
sleek wolf
ye
safe parrot
I multiplied with the wrong line
kinda hard to keep track when every video puts them in different locations
brb trying again lol
yaay i did it
x^2+2x+2
safe parrot
yes !
sleek wolf
try!
you might not be able to, however
in that case, the final answer to the factoring of x^3 -2x - 4 is (x - 2)(x^2 + 2x + 2)
sleek wolf
😎
flat frigateBOT
safe parrot
sleek wolf
🎉
bleak rose
Determine the values of a and b for which the system of equations has infinitely many solutions
safe radishBOT
Determine the values of a and b for which the system of equations has infinitely...
bleak rose
I've reduced it a bit and we find
why am i now when trying to solve for inft many solutions allowed to just say:
-2-3a = 0 and solve for a -> -2/3
and
-7-3b = 0 and solve for b -> -7/3
past birch
why am i now when trying to solve for inft many solutions allowed to just say:
-...
because then you'd have 2 equations with 3 unknowns? (bottom row is 0 0 0 | 0)
stoic dune
You're not done reducing
bleak rose
You're not done reducing
the answer can we found
stoic dune
If you continue reducing, you'll have to divide by -2 - 3a
But that's impossible if -2 - 3a = 0
bleak rose
but there will be inft many solutions for what i've wrote
and that is just because we get a row of 0's?
stoic dune
Let's say -2 - 3a = 0. Then the bottom row implies:
0 = -7 - 3b
But that can't happen unless -7 - 3b = 0 itself
bleak rose
yes
stoic dune
So there's no solution, if that's not the case
If -7 - 3b does equal 0, then we'd have infinite solutions
bleak rose
why dont we have that before
or rather why isnt that possible before that
iguess if it wanst 0
stoic dune
Which?
bleak rose
2 unknowns and 3 equations is surely solveable?
narrow mortar
For the system to have infinitely many solutions, the third row must represent a true statement when the system is consistent. This means that the third row should become (0x_1 + 0x_2 + 0x_3 = 0), which implies that (-3a+2 = 0) and (-3b-7 = 0).
flat frigateBOT
Flamey
stoic dune
Let's say -2 - 3a = 0
Let's say b = 1
Then the bottom row would read:
0 = -10
But that can never happen, no matter what x,y,z are.
bleak rose
you can have infitenly many solutions that dont cover all numbers or?
stoic dune
"cover all numbers"?
Let's say instead that
b = -7/3
Then the bottom row would read 0 = 0
And that's always possible. So this basically eliminates the bottom row. This way, you get infinitely many solutions
bleak rose
so it becomes linearly depedent?
stoic dune
When -2 - 3a = 0, the matrix has linearly dependent rows.
In that case, you can't have a unique solution anymore
bleak rose
because you can express the last row with anything * 0?
bleak rose
When -2 - 3a = 0, the matrix has linearly dependent rows.
i thought a matrix had infinitely many solutions when it has more varialbes than equations
if the number of variables is more than the number of nonzero rows in the reduced row-echelon form of the matrix
and here the variables are just x1,x2 and x3, so if we remove one row, we have 2 equations for 3 unknowns
which is obtained by making the last row full of zeroes, hence we make -2-3a/-7-3b =0
safe radishBOT
@bleak rose Has your question been resolved?
livid bronze
help (4b^2)^3
safe radishBOT
help (4b^2)^3
quasi bison
(4b^2)^3 is not equal to 64^6. the b does not disappear.
bleak rose
help (4b^2)^3
((xy)^n = x^n y^n)
flat frigateBOT
akiyama
bleak rose
$$(4b^2)^3 = 64b^6 $$
this will be your answer
bleak rose
the ^3 is applied to everything inside the parentheses
with that being said, could you scroll slighty up in the chat and see if what i wrote is indeed correct
flat frigateBOT
akiyama
safe radishBOT
@livid bronze Has your question been resolved?
mystic timber
safe radishBOT
mystic timber
The answer says that it is a but people are leaving steady and only once
Therefore the remaining visitors kicked out from the park my final answer is B
safe radishBOT
@mystic timber Has your question been resolved?
safe radishBOT
@mystic timber Has your question been resolved?
safe radishBOT
@mystic timber Has your question been resolved?
lean otter
safe radishBOT
lean otter
i dont really understand how the distance between P and L is |d + r cos theta|
if dist(F,L) = d then dist(F, L) = dist(P, F)(cos(theta)) + dist(P, L) ==> d = dist(P, F)(cos(theta)) + dist(P, L)
dist(P, F) = r then d = |r cos(theta)| + dist(P, L) ==> d - |r cos(theta)| = dist(P, L) ??
safe radishBOT
@lean otter Has your question been resolved?
lapis lintel
Complete the diagram for each step of the solution also lmk the bottom questions
safe radishBOT
Complete the diagram for each step of the solution also lmk the bottom questions
@lapis lintel Has your question been resolved?
lapis lintel
no
safe radishBOT
@lapis lintel Has your question been resolved?
safe radishBOT
@lapis lintel Has your question been resolved?
lapis lintel
Replace n by 2 you’ll see it does not work
Ok so explain please
like the other ones
brazen galleon
like the other ones
You don’t know how to solve equations at all ?
You don’t know when you switch something from side it changes the sign etc?
lapis lintel
You don’t know how to solve equations at all ?
why are you being mean bruh 😭 i came here for help
brazen galleon
why are you being mean bruh 😭 i came here for help
I’m not mean sorry if that’s how it felt
I was just asking
To know where I can help
lapis lintel
You don’t know how to solve equations at all ?
I do but not really like that
I can do like 3n + 7 or like 3n - 7
stuff like that
brazen galleon
I can do like 3n + 7 or like 3n - 7
For which line
lapis lintel
It was an example
brazen galleon
Ok lets take the first line
lapis lintel
Ok
brazen galleon
3n-4=5
You should always sépare n and numbers on a different side first
What happens if you do this here
lapis lintel
Okay but I need to draw it in the page and I have no idea how many to do
Or the bags
like that
brazen galleon
First one is done already I think
lapis lintel
Yes
For the second one
how do u draw it?
like how many should I do
on each side
i think left side is bags and right is like the marbles or counters or wtv
brazen galleon
I think you’re expected to put it just as the equation is written for each step
lapis lintel
Oh like simplifying ?
brazen galleon
For the second I’d say you put 3n-4+4 on left and 5+4 on right
lapis lintel
Okay what abt the next one
brazen galleon
Idk how you can draw the divide
lapis lintel
Man idk either
Istg we r supposed to use the bags and like draw it
Wait
Nvm the way u said it makes sense
Ok skip the third one what abt the last one
brazen galleon
Ok skip the third one what abt the last one
Just like you did
With 3 instead of 2
lapis lintel
With 3 instead of 2
Can u write it out
brazen galleon
Can u write it out
Just put a last marble
lapis lintel
Just put a last marble
not tryna be dumb or anything but i got ABSOLUTELY no idea what ur saying
I think u might tell math is not my strongest subject
brazen galleon
You drew 2 marbles yeah ?
brazen galleon
I think u might tell math is not my strongest subject
Fine 🤣
lapis lintel
You drew 2 marbles yeah ?
WHERE
lapis lintel
That’s not right
Someone told me
Ok so 9 marbles in the second right side
How do i turn the rest into the bags
@brazen galleon GET BACK ONLINE RN
brazen galleon
🤣
It’s 4am
I have mathematics in 4h
And I’m not going to draw marbles
😭😭
Put 3n bag on the left with the 9 then
lapis lintel
3 BAGS
brazen galleon
Is it that big of deal ? You understood the point
brazen galleon
What is my question
lapis lintel
What is my question
It’s ok i got my brothers gf to help u a real one fr tho for putting up w me
brazen galleon
Alright ahah nice
safe radishBOT
@lapis lintel Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
lean otter
<booktabs, multirow, colortbl, sansmathfonts>
\let\bm\boldsymbol
\setcolors{bg=2c3b42, fg=a7b8bf, orange=ba694b}
\begin{tabular}{cccccccccc}
\toprule
\multicolumn{2}{c}{\multirow[b]{2}{5em}[.5ex]{\centering\vfill\sffamily\bfseries Present State}} &&
\multicolumn{4}{c}{\sffamily\bfseries Next State} &&
\multicolumn{2}{c}{\sffamily\bfseries Output} \\ \cmidrule{4-7} \cmidrule{9-10}
\multicolumn{2}{c}{} &&
\multicolumn{2}{c}{$\bm {x = 0}$} & \multicolumn{2}{c}{$\bm {x = 1}$} &&
$\bm {x = 0}$ & $\bm {x = 1}$ \\ \cmidrule{1-2} \cmidrule{4-7} \cmidrule{9-10}
$\bm A$ & $\bm B$ && $\bm A$ & $\bm B$ & $\bm A$ & $\bm B$ && $\bm y$ & $\bm y$ \\
\arrayrulecolor{orange}\midrule \arrayrulecolor{fg}
$0$ & $0$ && $0$ & $0$ & $0$ & $1$ && $0$ & $0$ \\
$0$ & $1$ && $0$ & $0$ & $1$ & $1$ && $1$ & $0$ \\
$1$ & $0$ && $0$ & $0$ & $1$ & $0$ && $1$ & $0$ \\
$1$ & $1$ && $0$ & $0$ & $1$ & $0$ && $1$ & $0$ \\
\bottomrule
\end{tabular}
lean otter
for the above table (credits to snowo), i need to understand what my book means with
"The first 0 input after a string of 1’s gives an output of 1 and transfers the circuit back to the initial state, 00."
what does that sentence mean exactly
safe radishBOT
@lean otter Has your question been resolved?
safe radishBOT
@lean otter Has your question been resolved?
safe radishBOT
@lean otter Has your question been resolved?
toxic stratus
"The first 0 input after a string of 1’s gives an output of 1 and transfers the ...
what even is this table
lean otter
what even is this table
so this table is like
describing a finite state automaton
idk what else to say 
its what u call a state table
toxic stratus
hmmge
lean otter
8 rows
toxic stratus
why are states 10 and 11 even a thing 
anyway
isnt the sentence saying exactly what it means
like
after having received a whole bunch of consecutive 1s in the input
receiving a 0 will cause an output of 1
and x=0 always puts the state back into AB = 00
safe radishBOT
@lean otter Has your question been resolved?
toxic stratus
lean otter
ok but
like
i get that
but
u have to know that you are working with x = 1
for the thing to work yeah?
because x = 1 just tells u that they are 0's nevertheless
toxic stratus
wot are you saying
grim plover
what is each row representing?
lean otter
.close
safe radishBOT
safe radishBOT
Available help channel!
lean otter
safe radishBOT
lean otter
the way i solved A was i found the derivative of the function
and then the derivative for x = 75
A)
C'(x) = 3.6x + 54 - 0.2x^1.5
C'(75) = 3.6(75) + 54 - 0.2(75)^1.5 = 194.1
Is it correct?
lean otter
positive
frozen marlin
in this case
absolutely
and since the derivative is the slope--the rate of change--will it increase or decrease in the near future?
keeping in mind that it's positive
lean otter
Increase?
frozen marlin
yes!
since it's positive, it means that the profit's increasing--the slope is increasing, which means that it's increasing at that point
frozen marlin
okay
at the maximum, will the derivative be increasing or decreasing?
keep in mind that the profit NEVER goes higher than the maximum
lean otter
i mean its the maximum?