looks right
#help-49
buoyant yoke
midnight plankBOT
@rare wave Has your question been resolved?
rare wave
G is 0
H is 0
I is + infinity
O is ?
P is ?
buoyant yoke
G is 0
H is 0
I is + infinity
O is ?
P is ?
G, wrong
H, correct
I, correct,
buoyant yoke
<@&286206848099549185>
!15m
midnight plankBOT
<@&286206848099549185>
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
rare wave
This is easy
rare wave
Oh
buoyant yoke
left limit is 0
right limit is 5
thus limit DNE
and f(6)=0
so that one was right
but it was crossed out
so i wasnt sure
buoyant yoke
ly2 bro ❤️
rare wave
Im studying for my exam
buoyant yoke
and people will help
rare wave
ok✌️✌️
long bough
midnight plankBOT
@rare wave Has your question been resolved?
viral dagger
let there be positive reals $a_1,a_2,\dots,a_{19},b_1,b_2,\dots,b_{19}$, let $A=a_1+a_2+\dots+a_{19}$ and $B=b_1+b_2+\dots+b_{19}$. Prove that there exists a subset $S\subset{1,2,\dots,19}$ with $|S|=10$ such that $\sum_{i\in S}a_i>\frac{A}{2}$ and $\sum_{j\in S}b_j>\frac{B}{2}$
midnight plankBOT
let there be positive reals $a_1,a_2,\dots,a_{19},b_1,b_2,\dots,b_{19}$, let $A=...
grand pondBOT
skissue.in.a.teacup
viral dagger
bro what are these questions
exotic pelican
induction could work
jaunty canopy
let there be positive reals $a_1,a_2,\dots,a_{19},b_1,b_2,\dots,b_{19}$, let $A=...
is n meant to be 19 here?
cus otherwise it's not true lol, take a_1 = a_2 = ... = a_n = b_1 = ... = b_n, n >>19
exotic pelican
though I only see a way if there exist i and j such that a_i >= a_j and b_i >= b_j
viral dagger
ermm wait
ok im dumb
whoopsie daisies
let there be positive reals $a_1,a_2,\dots,a_{19},b_1,b_2,\dots,b_{19}$, let $A=a_1+a_2+\dots+a_{19}$ and $B=b_1+b_2+\dots+b_{19}$. Prove that there exists a subset $S\subset{1,2,\dots,19}$ with $|S|=10$ such that $\sum_{i\in S}a_i>\frac{A}{2}$ and $\sum_{j\in S}b_j>\frac{B}{2}$
grand pondBOT
skissue.in.a.teacup
jaunty canopy
let there be positive reals $a_1,a_2,\dots,a_{19},b_1,b_2,\dots,b_{19}$, let $A=...
is it > A/2 and B/2 or >= A/2 and >= B/2?
exotic pelican
$n = 2m-1, |S| = m$, induct on $m$. Take $i$ and $j$ such that $a_i \geq a_j$ and $b_i \geq b_j$, then by hypothesis, there exists $S'$, $|S'| = |S| - 1$ such that \ $\sum_{i \in S'} a_i > \frac{A-a_i-a_j}{2}$ and $\sum_{j \in S'} b_j > \frac{B-b_i-b_j}{2}$
viral dagger
is it > A/2 and B/2 or >= A/2 and >= B/2?
grand pondBOT
rbit
viral dagger
wouldnt the base case here be like 19?
exotic pelican
but then idk about the case when you can't choose i and j this way
jaunty canopy
**skissue.in.a.teacup**
if n=3 instead, we had 1,2,3
exotic pelican
no base case is m=1
exotic pelican
then we only have a single number, trivial
jaunty canopy
you can only get >=
exotic pelican
a_1 > a_1 / 2
jaunty canopy
you can only get >=
wait nvm ignore me
jaunty canopy
let there be positive reals $a_1,a_2,\dots,a_{19},b_1,b_2,\dots,b_{19}$, let $A=...
i did a similar question a while back
viral dagger
if n=3 instead, we had 1,2,3
say its 123 and 321 for worst case, 3+1>6/2=3, 1+3>6/2=3?
jaunty canopy
say its 123 and 321 for worst case, 3+1>6/2=3, 1+3>6/2=3?
yeah i realised
jaunty canopy
i did a similar question a while back
n = 99 and it was >= although if ur question is with > then i suspect the same proof works
but tbh i've forgotten how to do it so i'm just reminding myself what i did
but i do know my motivation was "if i can show that if i pick 10 of these numbers at random, if the probability that it's > A/2 is > 1/2 then we'd be done"
so i know my overall method was to pick S randomly and show that the probability S works for just a is > 1/2
cus we can finish by symmetry
viral dagger
**rbit**
i mean your taking some a_i from the sum right? what if the a_i chosen is the smallest and nothing is equal to it, so a_j wouldnt exist?
viral dagger
so i know my overall method was to pick S randomly and show that the probability...
who the hell was the first person to put probability in number theory bro 😭
viral dagger
eugh they are simmilar
exotic pelican
yes I'm making a little assumption, this wouldn't work in the exceptional case that
a_1 >= a_2 >= ... >= a_n and b_1 < b_2 < ... < b_n
jaunty canopy
iirc this problem has like a billion possible solutions
viral dagger
erm
jaunty canopy
induction is possible too i think
and i think there's also some IVT-type shenanigan u can do?
viral dagger
iirc this problem has like a billion possible solutions
name 10000
viral dagger
and i think there's also some IVT-type shenanigan u can do?
ivt? as in the intermediate value thinggy?
jaunty canopy
yeah
viral dagger
whats the simplest of em all
viral dagger
no source given
midnight plankBOT
@viral dagger Has your question been resolved?
jaunty canopy
n = 99 and it was >= although if ur question is with > then i suspect the same p...
ok yeah i've read through my old proof and it can be adapted to > rather than >=
jaunty canopy
but i do know my motivation was "if i can show that if i pick 10 of these number...
anyway that was my approach and the motivation behind my approach but there are non-probabilisitc methods
jaunty canopy
iirc it was the IVT-type soln
i think for this soln, you can do some sort of weird IVT where you think about some subset S with 10 elements
and then change that one by one
until you've flipped it at the end
viral dagger
wtf
jaunty canopy
the intuitive thing to do is to WLOG a_1 <= a_2 <= a_3 <= ... <= a_19
then consider the last 10 numbers
and slide that along to the front
but this doesn't quite work
idr how you actually do it
viral dagger
midnight plankBOT
midnight plankBOT
Available help channel!
granite cipher
midnight plankBOT
granite cipher
Is this now correct
dawn dagger
Is this now correct
yes, but don't use implication arrows, just use =
midnight plankBOT
midnight plankBOT
Available help channel!
granite cipher
midnight plankBOT
granite cipher
Why is it 6
granite cipher
Sorry
dawn dagger
-54xcos(3x) looks like you haven't applied product rule
tacit rose
You did the derivative wrong here with -54cos(3x)
granite cipher
Oh
tacit rose
Check your math again
tacit rose
Also what @adoniszgks said too, don't close your channel and open a new one if you haven't gotten help
You waited 7 minutes
midnight plankBOT
midnight plankBOT
Available help channel!
olive yew
How should I try to do this question
midnight plankBOT
How should I try to do this question
@olive yew Has your question been resolved?
severe oxide
@olive yew You still need help?
midnight plankBOT
@olive yew Has your question been resolved?
tidal turret
midnight plankBOT
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
midnight plankBOT
@tidal turret Has your question been resolved?
tidal turret
<@1187916384341078070> Has your question been resolved?
Pi is the plane that contains L1 and L2
the normal of the plane pi is orthogonal to all the vectors inside the plane
so normal of Pi is orthogonal to the direction vector of L1 and orthogonal to the direction vector of L2
Pi is not given
but direction of L3 is (1,1,0)x(0,-1,-2)
the result of that is the normal of Pi
and since L3 is orthogonal to Pi
direction of L3 is the same as the normal of Pi
because is a vector perpendicular to the plane
L3 : X = λ(-2,2,-1) + (a,b,c)
now since L3 intersects L1 and L3 intersects L2
we need to find the intersection of L1 and L2
if its non empty we use that for the parametric equation of L3
otherwise we continue thinking
L1 : a(1,1,0) + (0,0,2)
L1 : (a,a,2)
L2 : b(0,-1,-2) + (1,-2,-4)
L2 : (1,-b-2,-2b-4)
(a,a,2)=(1,-b-2,-2b-4)
- a = 1
- a = -b-2
- 2 = -2b-4
1 = -b -2
1 +2 = -b
b = -3
2 = -2(-3)-4
2 = 6 - 4
L3 : X = λ(-2,2,-1) + (1,1,2)
.solved
midnight plankBOT
midnight plankBOT
Available help channel!
cyan fiber
8. Consider the vectors a⃗ = î - 2ĵ + 3k̂ and b⃗ = 4î - 4ĵ + 7k̂. What is the vector perpendicular to both vectors?
a) -10î + 3ĵ + 4k̂
b) 10î - 3ĵ + 4k̂
c) -10î - 3ĵ + 4k̂
d) 10î - 3ĵ - 4k̂
midnight plankBOT
**8. Consider the vectors a⃗ = î - 2ĵ + 3k̂ and b⃗ = 4î - 4ĵ + 7k̂. What is t...
cyan fiber
How to do this one
heady plume
Look into the definition of Cross Product.
heady plume
Unfortunately, you'll see none of the given options are perpendicular to both a, b
visual tiger
indeed, no way to make vector a perpendicular to any of those propositions
their dot product is always $\pm 10 \pm 6 \pm 12$
grand pondBOT
rafilou is not not born in 2003
visual tiger
which is never 0 mod 3
visual tiger
OOOO SO BASICALLY THE CROSS PRODUCT
the question is flawed
cyan fiber
the question is flawed
Oh ow ic ic
delicate sage
Oh ow ic ic
The question must be
cyan fiber
their dot product is always $\pm 10 \pm 6 \pm 12$
Dot product?
delicate sage
Perpendicular to the difference of two vectors
cyan fiber
Why did u take dot product
delicate sage
Why did u take dot product
Dot product of perpendicular vectors yields 0
cyan fiber
Dot product of perpendicular vectors yields 0
OOOOOH
visual tiger
Dot product of perpendicular vectors yields 0
it was yields
cyan fiber
it was yields
😭 was?
granite cipher
midnight plankBOT
granite cipher
Why would this be wrong?
And how do u know when it's Cot-1 or tan -1 ?
Because the only difference is a negative sign
How do u know if its tan or cot
heady plume
Give me a minute
The error is not whether you wrote arccot or the ans key has arctan
The error is
fallow scarab
How do u know if its tan or cot
both are valid answers
heady plume
Also, arctan x + arccot x = π/2, so your integral is valid
heady plume
As long as you fix that denom 3
granite cipher
It's not a test
Otherwise it wouldn't tell me if I was wrong...
Its practice questions for my exam tomorrow
dusty portal
granite cipher
I'm stuck
granite cipher
<:honestgiantreaction:1291055542252339200>
😂😂
dusty portal
you did synthetic division, right?
granite cipher
Idk what that is but my method is here
fallow scarab
do you need parentheses around your x/sqrt(8)
at this point it's just a formatting or software problem and nothing to do with math
dusty portal
do you need parentheses around your x/sqrt(8)
that is the correct answers format
drifting birch
Factoring out 3 but not simplifying √8😂
dusty portal
I've got am exam tomorrow and they use this stupid software
it’s the same thing 😭😭😭
bad software imo
pine thicket
do you get to see the correct answers and your answers after the test
dusty portal
so de it’s correct
signal flare
oh dear god these are the worst
granite cipher
do you get to see the correct answers and your answers after the test
Nop going out blind and praying to god I formatted right
dusty portal
ah wait
granite cipher
Oh
granite cipher
No probs
granite cipher
Thanks guys u saved my sanity 🙏
drifting birch
Nop going out blind and praying to god I formatted right
Ask your prof as to how it should be formatted
dusty portal
lol
pine thicket
if the software marks it wrong, but youre correct with the exception of your formatting, i would ask your teacher to correct it
granite cipher
Okay I’ll ask the professors
dusty portal
Thanks guys u saved my sanity 🙏
you can also argue with the teacher and present the issue
prof will probably accept it
pine thicket
much easier to appeal to the human than the machine
granite cipher
Yea definitely
dusty portal
midnight plankBOT
dusty portal
https://tenor.com/view/clash-royale-goblin-laugh-gif-13901438
u play clash?
dusty portal
oh nice
granite cipher
Do u?
dusty portal
i peaked at top 5k
granite cipher
i peaked at top 5k
Goated man
granite cipher
Did u spend much?
dusty portal
top ladder is too hard to grind
granite cipher
i quit since
It’s sooo p2w😭
dusty portal
Did u spend much?
you bet...
granite cipher
top ladder is too hard to grind
Honestly
I can’t even get 9k I’m drowning at 7.8k 😭
granite cipher
you bet...
😂😂😂
dusty portal
I can’t even get 9k I’m drowning at 7.8k 😭
lol
granite cipher
Logbait is cooked now
dusty portal
i never finished 9k cause i stopped grinding
granite cipher
i never finished 9k cause i stopped grinding
Ahh will u ever come back?
dusty portal
i’m at like 8.6k trophies now
granite cipher
i’m at like 8.6k trophies now
You could get 9k
granite cipher
it’s a hellhole for me
Prob a good thing u quit it then
I don’t play much clash royale but I play clash of clans now
I started recently tho
U should give clash of clans a try if u haven’t played that
midnight plankBOT
u play clash?
This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.
Available help channel!
split tundra
Hi
split tundra
Need help with this
I proved it as one-one
But the onto portion is a bit confusing,
I got it (it's not onto)
But the answer says othervise
dreamy lichen
I got it (it's not onto)
Why is it not?
sharp coral
it defines R+ as the nonnegative real numbers
dreamy lichen
Which isn't in our domain
Read the defn of R+
split tundra
Because R^+ means >0
dreamy lichen
0 isn't negative afaik
dreamy lichen
Because R^+ means >0
that's not what it says
I'd probably interpret it as such, but the question explicitly states a different definition
so you should use that one
pine thicket
also that would disprove one-to-one/inrectivity, not it being onto/surjective
grim vector
0 isn't negative afaik
Depends convention i believe
pine thicket
if it were in the domain
grim vector
But not the point here
dreamy lichen
Depends convention i believe
Huh
dreamy lichen
No but it's true R+is >0
dreamy lichen
But 0 isn't in that !?
0 isn't negative (at least using the definitions im aware of)
dreamy lichen
I mean zero isn't -ve nor its positive
Exactly
0 isn't negative
hence 0 is non-negative
non-negative isn't the same as positive
we say "non-negative" for the sole reason of including 0
gusty falcon
Huh
i’ve heard some people treat 0 as both positive/negative (the french lol) and some people even treat 0 as prime
radiant roost
midnight plankBOT
radiant roost
i finished exercise 16 but it would be nice if someone could check it
i'll try exercise 17 after lunch but it looks even worse than 16 
here's my answers
oh yeah, i never used 14(b) so i think i missed something
midnight plankBOT
@radiant roost Has your question been resolved?
midnight plankBOT
@radiant roost Has your question been resolved?
midnight plankBOT
@radiant roost Has your question been resolved?
midnight plankBOT
@radiant roost Has your question been resolved?
midnight plankBOT
@radiant roost Has your question been resolved?
radiant roost
.close
prime swallow
When you complete the square do you have to solve for a? In a quadratic equation
midnight plankBOT
When you complete the square do you have to solve for a? In a quadratic equation
fallow scarab
When you complete the square do you have to solve for a? In a quadratic equation
Depends on the question
prime swallow
Do you only solve for a if you find the equation?
This is finding the intercepts?
fallow scarab
No you don't need to find the leading coefficient
rich sparrow
lite work
midnight plankBOT
@prime swallow Has your question been resolved?
vernal bane
If you wanna use the quadratic equation, you gotta rearrange and then solve for x
prime swallow
My school calls it the quadratic formula
tame forge
-4.93
vernal bane
Yeah, same difference, main thin is you gotta make it so f(x)=0
tame forge
My school calls it the quadratic formula
fancy stuff
vernal bane
But it looks like you can try factorising
tacit rose
But it looks like you can try factorising
If you re-read the instructions in that image, it says solve using complete the square
Not by factoring
Or quadratic formula
prime swallow
What is it called when you know wx is the same as vx? Is it just called isoscles triangles?
prime swallow
How do you know wx is equal to vx? Is it just because of isoscles triangles?
midnight plankBOT
How do you know wx is equal to vx? Is it just because of isoscles triangles?
@prime swallow Has your question been resolved?
midnight plankBOT
rough lance
How do you know wx is equal to vx? Is it just because of isoscles triangles?
Yes
Isosceles triangle have 2 sides equal
midnight plankBOT
Available help channel!
gusty falcon
How many ways can 5 people be assigned to 3 distinct teams, with at least one team having 2 members?
midnight plankBOT
How many ways can 5 people be assigned to 3 distinct teams, with at least one te...
gusty falcon
Cases:
-
(2,2,1) and all permutations of this
-
(3,1,1,) and all permutations of this
-
(4,0,1) and all permutations of this
-
(5, 0, 0) and all permutations of this
does the question suggest that we care about case 3. and case 4.?
rugged saffron
I assume the last two would be 2 and 1 distinct teams respectively, so I would say no we don’t
gusty falcon
I assume the last two would be 2 and 1 distinct teams respectively, so I would s...
let's suppose that you already have predetermined the 3 "distinct" teams
A B and C
then now the question is how many ways can you assign these 5 people into 3 teams
with the obvious restriction (at least one time bla bla)
so like technically nothing is stopping me from just assigning all 5 to one team
no?
also if we change the "people" to "objects" and "distinct teams" to "distinct bins" then it's not far fetched now to say that u can have 5 objects in one bin and the other 2 bins have no objects
rugged saffron
I suppose when you talk about abstract objects, but for me it does not make sense for the specific example. Also, in the question isn’t the additional constraint that one team has at least 2 members just redundant?
gusty falcon
I suppose when you talk about abstract objects, but for me it does not make sens...
what about in my new question
with objects and distinct bins
rugged saffron
Yeah then there is no familiar constraint about each bin to be nonempty to be classed as a bin in the first place
gusty falcon
wdym
midnight plankBOT
@gusty falcon Has your question been resolved?
rugged saffron
As in when we talk about teams, implicitly there is at least one person for it to be a team, otherwise it just doesn’t really exists
Team is a sort of social construct rather than a room e.g.
trail parrot
differentiation of 1/(b-x)² , b is constant.
will the answer be -1×(b²+x²-2bx)×(2x-2b)
i opened square identity in denominator, and then put it in numerator with power -1 and then differentiated as normal.
is this right?
midnight plankBOT
differentiation of 1/(b-x)² , b is constant.
will the answer be -1×(b²+x²-2bx)...
trail parrot
my attempt
midnight plankBOT
@trail parrot Has your question been resolved?
trail parrot
<@&286206848099549185>
i checked and my solution is wrong. pls someone help me how to do it
i solved it
i have another question.
V=2x+4y , find magnitude at (1.3)
if there's a 2 dimensional vector, then it's magnitude will be √{(x component)²+(y component)²} = √13
V=2x at x=2
but if it's a one dimensional vector, then what will be its magnitude? will it be the vector itself = 4
vast pulsar
differentiation of 1/(b-x)² , b is constant.
will the answer be -1×(b²+x²-2bx)...
No, you're missing something.
trail parrot
i figured it out
trail parrot
i have another question.
V=2x+4y , find magnitude at (1.3)
if there's a 2 dime...
i wrote second question
vast pulsar
i have another question.
V=2x+4y , find magnitude at (1.3)
if there's a 2 dime...
I don't know much about vectors, sorry...
trail parrot
i have another question.
V=2x+4y , find magnitude at (1.3)
if there's a 2 dime...
<@&286206848099549185>
midnight plankBOT
@trail parrot Has your question been resolved?
tepid zephyr
Hello I need help
midnight plankBOT
Hello I need help
buoyant yoke
!15m
midnight plankBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
tepid zephyr
I just want to know where 2y came from
buoyant yoke
i think they added the second row to the first row
but then it should be 2 lambda
so they mightve made a mistake?
tepid zephyr
i think they added the second row to the first row
If they did then the second row should disappear but it's still there
buoyant yoke
If they did then the second row should disappear but it's still there
no it doesnt
its a system of equations
have you had linear algebra?
tepid zephyr
have you had linear algebra?
Yeah
tepid zephyr
its a system of equations
Right
buoyant yoke
$\left(\begin{array}{cc|c}2&0&\lambda\0&2&\lambda\1&1&1\end{array}\right)$
grand pondBOT
Bonk
tepid zephyr
I think they substituted lambda in the first equation by the second row which is gonna be 2x = -2y + lambda and then they switched 2y next to 2x which is gonna give 2x + 2y = lambda
I don't know what I'm saying
buoyant yoke
how are you getting 2x=-2y+lambda?
tepid zephyr
It just makes sense in my brain
By switch 2y = lambda by 0 = -2y + lambda
But that's not right
Right ?
buoyant yoke
**Bonk**
$\left(\begin{array}{cc|c}2&0&\lambda\0&2&\lambda\1&1&1\end{array}\right)\sim\left(\begin{array}{cc|c}2&2&2\lambda\0&2&\lambda\1&1&1\end{array}\right)\sim\left(\begin{array}{cc|c}1&1&\lambda\0&2&\lambda\1&1&1\end{array}\right)$
grand pondBOT
Bonk
buoyant yoke
the row 1 and row 3 give lambda = 1
tepid zephyr
How
buoyant yoke
do you still remember row manipulation from linear algebra?
tepid zephyr
A little bit
buoyant yoke
**Bonk**
here, i take row 2, and add it to row 1, then row 1 i divide everything by 2
tepid zephyr
Oh okay I understand
So I think they were supposed to write 2lambda
Then divide the row by 2
2 = lambda
Right ?
Now what happened in here???
buoyant yoke
they skipped a few steps
but yeah, they made a mistake
since they get lambda=2
then 2y=2 which gives y=1
but they say y=1/2
tepid zephyr
It's so confusing
tepid zephyr
Ok okay
buoyant yoke
**Bonk**
just like here
$\left(\begin{array}{cc|c}2&0&\lambda\0&2&\lambda\1&1&1\end{array}\right)\sim\left(\begin{array}{cc|c}2&2&2\lambda\0&2&\lambda\1&1&1\end{array}\right)\sim\left(\begin{array}{cc|c}1&1&\lambda\0&2&\lambda\1&1&1\end{array}\right)\sim\left(\begin{array}{cc|c}1&1&\lambda\0&2&\lambda\0&0&1-\lambda\end{array}\right)\sim\left(\begin{array}{cc|c}1&0&\frac{\lambda}{2}\0&1&\frac{\lambda}{2}\0&0&1-\lambda\end{array}\right)\implies\begin{cases}\lambda=1\x=\frac12\y=\frac12\end{cases}$
grand pondBOT
Bonk
buoyant yoke
that looks kinda ugly...
tepid zephyr
How do u even read this
tepid zephyr
Yeah a little
tepid zephyr
It doesn't show any steps
buoyant yoke
okay
tepid zephyr
Wait
buoyant yoke
take your time
buoyant yoke
they added an extra one
tepid zephyr
That's possible?
buoyant yoke
its some algebraic manipulation
buoyant yoke
That's possible?
well, its simply a systems of equations
so you can add and remove any equation you like
but here, that top row forms from the 2nd and 3rd rows
consider this:
\begin{align*}a=a\b=b\end{align*}
grand pondBOT
Bonk
buoyant yoke
now, that top row, i multiply both sides by 1
and thats allowed cuz i dont change anything
giving: $a=a\iff a\cdot 1=a\cdot 1$
grand pondBOT
Bonk
buoyant yoke
yes?
tepid zephyr
Yes
buoyant yoke
now, we can transform that 1 into a b/b
since a number divided by itself is 1
$a\cdot 1=a\cdot 1\iff a\frac{b}{b}=a\frac{b}{b}$
grand pondBOT
Bonk
grand pondBOT
Bonk
buoyant yoke
$a\frac{b}{b}=a\frac{b}{b}\iff ab^2=ab^2$
grand pondBOT
Bonk
buoyant yoke
this is kinda basic algebraic manipulation
we can also go from one step: $a=a\implies ab=ab$
grand pondBOT
Bonk
grand pondBOT
Bonk
buoyant yoke
is also true
so if we have something like $4=7-3$ and $-4=-9+5$ then we can get $4\cdot(-4)=(7-3)(-9+5)$
grand pondBOT
Bonk
buoyant yoke
so in a similar vein we have
$\left.\begin{aligned}\frac{\sqrt{y}}{x}&=2\lambda\\frac{\sqrt{x}}{y}&=3\lambda\end{aligned}\right}\implies\frac{\sqrt{y}}{x}\frac{\sqrt{x}}{y}=2\lambda\cdot 3\lambda$
grand pondBOT
Bonk
buoyant yoke
which is precisely the 1st row
tepid zephyr
Yeah I understand that
Thank you so much
Can I ask u again?
What happened in here
tepid zephyr
they squared both sides
Yeah that's what I was thinking until I saw lambda disappear
buoyant yoke
\begin{align*}\sqrt{y}&=2\lambda\sqrt{x}\\left(\sqrt{y}\right)^2&=(2\lambda\sqrt{x})^2\y&=4\lambda^2x\end{align*}
grand pondBOT
Bonk
buoyant yoke
Yeah that's what I was thinking until I saw lambda disappear
that lambda disappears because they fill it in
tepid zephyr
Yes I understand now
buoyant yoke
great
tepid zephyr
Wait
buoyant yoke
waiting...
tepid zephyr
Don't go yet wait
tepid zephyr
dw, im still here
Thank you
grand pondBOT
Bonk
buoyant yoke
to find these, you need to plug y in to x or vice versa
tepid zephyr
It's gonna give you 1
grand pondBOT
Bonk
buoyant yoke
dont plug in x here
buoyant yoke
**Bonk**
this has one variable
you can just find x here
\begin{align*}2x+3\frac23x&=10\2x+2x&=10\4x&=10\end{align*}
grand pondBOT
Bonk
tepid zephyr
buoyant yoke
honestly, this should be kinda basic 😅
buoyant yoke
this is uni level right?
tepid zephyr
Yeah
buoyant yoke
which year?
tepid zephyr
First year
I study international economics so this is like basic math compared to other majors
tepid zephyr
that you werent a math major
I was an art major 😭
buoyant yoke
why did you swap?
tepid zephyr
I just like math that's why I know a little bit
buoyant yoke
fair enough 😂
tepid zephyr
That was such a stupid decision
If I had chosen maths i wouldn't have these kind of problems now
buoyant yoke
i feel like most people that are rly into art dont have to do some kind of art major
they just kinda do it
tepid zephyr
i feel like most people that are rly into art dont have to do some kind of art m...
Yeah true
I don't even know why I did art
It's fun and easy yeah
But it's useless
What are you gonna with an art major???
tepid zephyr
It's literally worse than communications
tepid zephyr
make paintings and shit
AI does that now
tepid zephyr
cant make anything original
Yeah true
But I still don't think an art major is really gonna get your far
You*
buoyant yoke
it might give connections
and show you your work to others that might otherwise not see it
but idk the art world at all
buoyant yoke
thats unfortunate
but yeah, if youre struggling with this algebra stuff
id recommend practicing it
khanacademy might be a good resource for oyu
do you have any more questions @tepid zephyr ?
!done
midnight plankBOT
If you are done with this channel, please mark your problem as solved by typing .close
tepid zephyr
khanacademy might be a good resource for oyu
Yeah I do use that
midnight plankBOT
midnight plankBOT
Available help channel!
gusty falcon
when throwing one fair dice, it's not "mutually exclusive" right?
midnight plankBOT
when throwing one fair dice, it's not "mutually exclusive" right?
gusty falcon
for mutual exlusivity you need at least two events, no? 😭
idk why this is written, it's wrong right?
urban widget
Mutually exclusive means two or more things cannot happen at the same time.
gaunt jetty
for mutual exlusivity you need at least two events, no? 😭
an "event" doesn't imply what actually happens. It is a collection of the possible outcomes that are a part of the sample space
hard umbra
the events {1} and {2} are mutually exclusive
last arch
for mutual exlusivity you need at least two events, no? 😭
I mean technically they are
Rolling a 1 and rolling a 2 are two events
urban widget
You’re probably thinking about mutually inclusive events.
gusty falcon
idk what they mean by 3)
gusty falcon
You’re probably thinking about mutually inclusive events.
hmm
last arch
Within two dice rolls it's possible to get a 1 and 2
hard umbra
the events are now more complicated
last arch
Within two dice rolls it's possible to get a 1 and 2
So they're not mutually exclusive events any more
hard umbra
,, \set {\text{getting a 1}} = \set {(1, x) \where 1 \le x \le 6} \union \set {(x, 1) \where 1 \le x \le 6} \
\set {\text{getting a 2}} = \set {(2, x) \where 1 \le x \le 6} \union \set {(x, 2) \where 1 \le x \le 6}
grand pondBOT
hard umbra
these two events are not mutually exclusive
gusty falcon
😭 i'm confused so all those 3 claims are right
but the reasoning is a bit handwavy?
gusty falcon
,, \set {\text{getting a 1}} = \set {(1, x) \where 1 \le x \le 6} \union \set {(...
yeah this makes sense righttt
hard umbra
i dont think giving rigorous reasoning was the intent
last arch
I mean it's just a simplified explanation
hard umbra
they are providing intuitive explanations which suffice
gusty falcon
yeah but then they conclude by saying "throwing two dices once is different than throwing one dice twice."
owing to those 3 points
i don't see the link
hard umbra
do they say that?
gusty falcon
yes right after
hard umbra
wild
urban widget
Huh
hard umbra
you should ask what they mean by "different"
last arch
Just ignore the last part
gusty falcon
you should ask what they mean by "different"
okay just to make things "clear"
in terms of independence
they're actually the "same" right?
hard umbra
well it matters what you care about
gusty falcon
using the joint probability defn of independence maybe coincides with just the product of two events occuring?
hard umbra
like obviously throwing two dice is not the same as throwing one die twice in a very strict sense of "same"
last arch
What's the source anyway
hard umbra
but if you only care about the two numbers that resulted, then they produce the same sample space and probabilities
last arch
What's the source anyway
Of this paragraph
gusty falcon
What's the source anyway
😭 it's not an "official source"
that's what my teacher had to say
gusty falcon
but if you only care about the two numbers that resulted, then they produce the ...
wwdym two numbers that resulted?
hard umbra
perhaps your teacher should hear what you have to say...
blissful totem
the last part is goofy
last arch
Why are you even reading this anyway😭
hard umbra
wwdym two numbers that resulted?
whether you throw two dice, or throw one die twice, you can record down two numbers
gusty falcon
whether you throw two dice, or throw one die twice, you can record down two numb...
correct
last arch
I'm sure there are more rigorous explanations out there you can find easily
hard umbra
if those two numbers are the only things you care about, then they are the same
gusty falcon
I'm sure there are more rigorous explanations out there you can find easily
well
idk i just read what he has to say as well
or that'd be disrespectful
😭
gusty falcon
if those two numbers are the only things you care about, then they are the same
yeah that makes sense 👍
last arch
Welp he deserves the disrespec-
gusty falcon
wait so do we see the link of how those 3 points links with "Can two mutually exclusive events be independent?"
according to me the answer is just yes
cuz P(A n B) = P(A) * P(B)
and well the product can be 0
gusty falcon
but i guess he went the route of using those 3 points
so idk what the implication is
consider A happening with a probability of 1 and B happening with a probability of 0
this is a mutually exclusive event?
and the product is 0
hard umbra
are your notes trying to explain that mutually exclusive and independent are distinct concepts?
gusty falcon
are your notes trying to explain that mutually exclusive and independent are dis...
no it was trying to answer that question in specific
hard umbra
certainly its possible to have mutually exclusive and independent events
but as you have found, it requires the probability of one of the events to be 0
which is not very exciting
gusty falcon
are your notes trying to explain that mutually exclusive and independent are dis...
they're different right?
hard umbra
a probability 0 event is always independent of any other event
gusty falcon
the intersection is 0 for mutually exclusive and well the intersection for independent events is the product of said events
hard umbra
they're different right?
yes
gusty falcon
a probability 0 event is always independent of any other event
rightt
hard umbra
usually one only cares about "pairwise mutually exclusive"
gusty falcon
Welp he deserves the disrespec-
that's not nice! 😭
hard umbra
whereas the notion of independence for more than 2 events is slightly more complicated
gusty falcon
isn't it just the product of all individual probabilities
gusty falcon
mhm
hard umbra
P(A \cap B) = P(A) P(B)
P(A \cap C) = P(A) P(C)
P(B \cap C) = P(B) P(C)
P(A \cap B \cap C) = P(A) P(B) P(C)
gusty falcon
ohh
okay wait then what i'm saying is for
mutually independence?
the one where we only care about $P(A \cap B \cap C) = P(A) P(B) P(C)$
grand pondBOT
mmmm7
gusty falcon
for 3 events, for example
hard umbra
independence for >2 events requires every single combination
theres not really a weaker notion
gusty falcon
i see
ughhhh wait 😭 i have to go
brb in 5- 10 mins
thanks for the help for now
hard umbra
midnight plankBOT
@gusty falcon Has your question been resolved?
tranquil zinc
is there somone who understand frensh
cunning fulcrum
how to find all homomoprhisms from Z[x]/(x^2)->Z24
midnight plankBOT
how to find all homomoprhisms from Z[x]/(x^2)->Z24
midnight plankBOT
@cunning fulcrum Has your question been resolved?
cunning fulcrum
2k, 3k, nk
runic hamlet
wait
you just said homomorphism
group or ring?
I guess ring cause thats more interesting
what can phi(x) be
cunning fulcrum
group or ring?
it doesnt work the same way?
midnight plankBOT
it doesnt work the same way?
runic hamlet
ring homomorphisms have the extra condition phi(xy)=phi(x)phi(y)
and depending on def also phi(1)=1
cunning fulcrum
ring homomorphisms have the extra condition phi(xy)=phi(x)phi(y)
oh yeah, so its gonna be k^2, k^3, k^n
midnight plankBOT
@cunning fulcrum Has your question been resolved?
runic hamlet
you have to remember that x^2=0
so 0=phi(0)=phi(x^2)=phi(x)^2
so what can phi(x) be?
cunning fulcrum
so 0=phi(0)=phi(x^2)=phi(x)^2
just 0
runic hamlet
why ?
midnight plankBOT
@cunning fulcrum Has your question been resolved?
wheat notch
.close
midnight plankBOT
prime hornet
midnight plankBOT
Available help channel!
gritty hatch
gritty hatch
ill never get off this problem smh
alr
can someone explain the
a^2/4
formula?
smoky solar
can you use parantheses to clarify it a bit?
gritty hatch
also dont be gestalt please sigh (explaining things with advanced wholes without explaining their parts)
gritty hatch
can you use parantheses to clarify it a bit?
ok so let me explain
5:13
ah yes cut 5 in half and square
that makes zero sense tbh
smoky solar
ok listen
slender walrus
would you be able to expand
(x + b/2)^2
gritty hatch
no because i dont even know what it means
i dont understand mathematical terminology
at all
gritty hatch
https://www.youtube.com/watch?v=P8W2M0jq2Qs&list=PLybg94GvOJ9FoGQeUMFZ4SWZsr30jl...
like i can read it in context of the video because like i understand what the cideo was saying
but when you show me a completely random term
it shouldnt even be possible for me to interpret that
because it has its own concept associated with it which is unknown to me
slender walrus
what random term
gritty hatch
would you be able to expand
(x + b/2)^2
this
smoky solar
it isn't random if you know what's going on
slender walrus
which term in there is random
gritty hatch
it isn't random if you know what's going on
i do not yet i do simutaneously
its the wording which i simutaneously understand and dont thats the issue
smoky solar
for example?
gritty hatch
for example?
because youd probably say like 1 thing and then id understand the entire sentence bc youd explain the goal of the operation but without the goal its meaningless to me because i cant apply the meaning behind the semantics to it
its gestalt
gritty hatch
would you be able to expand
(x + b/2)^2
see because even if i could read that i could interpret b/2 completely differently to b/2^2 which is what I specified at the very beginning i didnt understand so i understood but the phrasing makes it so confusing i forget
and i could interpret that as a wholly different equation
slender walrus
the things leas to end goal
things are explained one step at a time to ensure you understand each part
if you understand each part, there shouldn't be an issue with the conclusion
smoky solar
sometimes tou gotta follow till the last part to understand WHY we did some step that was seemingly random looking before
gritty hatch
would you be able to expand
(x + b/2)^2
i dont know it would take a while
slender walrus
take all the time you need
gritty hatch
talk about painful this will take time
smh
wiat
no that is literally hell
i cant do that
it would take like 1 hour
because the B term is a fraction which is also being squared meaning i cant square that and multiply it by itself because itd just take so long and i wouldnt know what the purpose of that is
slender walrus
indulge me
let's say this came up as a question
don't worry about anything else
and just do the expansion
it'll become clear why I'm asking you to do this after you do it
slender walrus
don't worry about fully simplifying
gritty hatch
(x + b/2)^2
tight osprey
bro what are u crying over
gritty hatch
x^2, xb, (b/2)^2, b/2 times x (i didnt know what side to put it on)
is that correct
slender walrus
make sure you have () around the (b/2)
tight osprey
huuh, i have my exam tomorrow and i haven't studied a bit
gritty hatch
huuh, i have my exam tomorrow and i haven't studied a bit
._.
slender walrus
x is your variable,so usually you'd want the constant coefficient at the front, bx
(x + b/2)^2 expands out to
x^2 + bx + (b/2)^2
do you have any issues with that so far?
midnight plankBOT
@gritty hatch Has your question been resolved?
midnight plankBOT
midnight plankBOT
Available help channel!
gritty hatch
(x + b/2)^2 expands out to
x^2 + bx + (b/2)^2
hm...
gritty hatch
that took... a while...
tbf i had to leave
shrug
also I was thinking about how people are and why im angry at people alot of the time and tbh i dont think people can help themselves and have psychological reasons for what they do so being angry at them is like being angry at the air, it just is, it isnt personal shrug.
so hopefully i can be more precise moving forward
sorry if thats a side tangent
anyways
as for your question
x^2 + bx + (b/2)^2```
because
xx = x^2 Left Left
x(b/2) Lright
(b/2)x RLeft
and
(b^2)(b^2) RightRight
(b^2)^2
yes
now the idea of using completing the square to solve quadratics is to reach an equation in the form
(something with x)^2 = constant
which you could then apply square roots to solve for x
which you should see in examples
gritty hatch
ikr
slender walrus
having
x^2 + bx present
you can introduce the (b/2)^2 term
(keep the equation balanced by adding that to both sides)
to get x^2 + bx + (b/2)^2 present
which you've just demonstrated is equal to (x + b/2)^2
gritty hatch
having
x^2 + bx present
you can introduce the (b/2)^2 term
(keep the equation b...
?
But that is seemingly irrelevent and not what I was asking from the beginning.
Sure, you can add it, and you can run numbers to show its equal and solve for X.
The problem is that (b/2)^2 does not exist.
You cant get it because it doesnt make sense.
How would you know (b/2)^2 specefically out of all possible numbers or operations would be the one number thatd have two factors that square to make itself and the B term?
slender walrus
because of the expansion you did
to get
x^2 + bx + (b/2)^2present
which you've just demonstrated is equal to (x + b/2)^2
the process revolves around that fact
slender walrus
how so?
gritty hatch
because how can it be equal to NaN?
slender walrus
where's NaN coming from
gritty hatch
slender walrus
where's NaN coming from
gritty hatch
You can only get (x+b/2)^2 if you already determined that (b/2)^2 is a perfect square factor to bx
which you cannot do
⚙️
gritty hatch
no no im saying that is the entire question
slender walrus
refer back to the task i gave you earlier of expanding
(x+b/2)^2
gritty hatch
as in you literally cannot do this without that
you cant!
you cant because (b/2)^2 isnt there
slender walrus
and you had no issue with expanding that out to x^2 + bx + (b/2)^2
forget all that,
don't get into chicken and egg stuff of what comes first
gritty hatch
i didnt because you already presupposed that (b/2)^2 was there when it wasnt
spare seal
Are you trying to understand how people came up with completing the square in the first place?
slender walrus
the idea is to manipulate what you have into something you know
spare seal
Or are you not convinced it is valid?
last arch
You can only get (x+b/2)^2 if you already determined that (b/2)^2 is a perfect s...
Are you sleepy right now
Maybe you should get some rest
gritty hatch
Or are you not convinced it is valid?
look
explain this to me
because this is outright impossible if this step isnt here
the only reason
they were even able to get the completed square
was cuz they said that
well they had
x^2 + 5x = 8/3
and they needed a number which would allow them to factor this which they did not have
so
they took 5x and they just forged a number
they took 5 and divided it by 2 to make 5/2 and powered it to make 25/2 or (5/2)^2
do you see why this doesnt work
last arch
I mean it's counter intuitive but i mean it does work
gritty hatch
no no i hate that that thing is the bane of my existence
slender walrus
you know
$$\violet{\br{x+\frac b2}^2} = \red{x^2 + bx} + \blue{\br{\frac b2}^2}$$
if you start with
$$\red{x^2+bx} = k$$
based on what you know, you can introduce that blue term
$$\red{x^2 + bx} + \underline{\blue{\br{\frac b2}^2}} = k + \underline{\blue{\br{\frac b2}^2}}$$
which helps you since you know the left side is now the same as the $\violet{\text{purple}}$
grand pondBOT
ℝαμOmeganato5
gritty hatch
i cant understand a single thing going on in the visual proof because they used the most overcomplicated semantics to make things i understood look incomprehensible
gritty hatch
because HOW do you know dividing and taking it to the power would do that?
slender walrus
you can introduce any term you want,
but the blue term is what you want due to that expansion/factorisation
gritty hatch
5 pm
slender walrus
wdym
because of the very first line/premise/expansion
that i asked you to do, its a very common expansion that you should know
just drill that into your mind
gritty hatch
the factors say 5 and 5 must be equal so when they are multiplied by the original number theyd not only come out to the original factor of 25 but also add to make 10 this is what completing the square is
last arch
(a+b)^2=a^2+2ab+b^2 do you believe this statement.
gritty hatch
but thats impossible because how are you going to f i n d a number with those properties in the first place?
slender walrus
is your issue with the presence of fractions?
gritty hatch
how do you find a number that both has factors that multiply to itself and add to the number we are trying to recreate
they used this bs method where they divided and squared it
but that makes no sense
slender walrus
yes
gritty hatch
nor is that true for all numbers
last arch
they used this bs method where they divided and squared it
I mean it's clearly not bs
gritty hatch
maybe?????????
last arch
Maybe?
slender walrus
show me a number where that isn't true
last arch
Tf you mean maybe
gritty hatch
100 div2
50 pow2 = .... BIG NUMBER
X50 = 50X
50X = 50X
gritty hatch
that still doesnt answer the question
slender walrus
you can leave it as 50^2
last arch
100 div2
50 pow2 = .... BIG NUMBER
X50 = 50X
50X = 50X
100 divided by 2 is 50
50+50=100
slender walrus
answer what question
slender walrus
do you 100% agree that
(x+b/2)^2 = x^2 + bx + (b/2)^2
gritty hatch
the thing is
is that
you are closer to speaking english as of rn
unlike many others
which was the source of all my anger
shrug
slender walrus
we've all be saying the same thing