no its how you present it
#help-49
gritty hatch
last arch
Answer this question here fr first
gritty hatch
Answer this question here fr first
possibly irrelevent
thinking about what you said earlier is more important rn
last arch
Do you actually believe this method is bs
Or is it just so abstract to you it's like bs
If its the former you should discard that notion for now
slender walrus
in the past, many people seemed to get annoyed because you kept dismissing what they said
gritty hatch
Do you actually believe this method is bs
I dont, I dont speak in the way you think
gritty hatch
in the past, many people seemed to get annoyed because you kept dismissing what ...
because they expressed it in a way that was too abstract
slender walrus
ok. forget about that then
gritty hatch
and refused to properly elaborate without increasing the ammount of abstraction
slender walrus
and just focus on what i asked you
gritty hatch
just trying to remember what you said earlier
you said any number times it
where
was it
slender walrus
do you 100% agree that
(x+b/2)^2 = x^2 + bx + (b/2)^2
slender walrus
$\frac n2 + \frac n2 = \frac{n+n}{2} = \frac{2n}{2} = n$
grand pondBOT
ℝαμOmeganato5
slender walrus
this is exactly what I didnt want to happen
didn't want what to happen
last arch
And then we smush those two halves together
gritty hatch
you said if you cut it in half and you applied a power to it it...
last arch
How much of the original pie do we have left
gritty hatch
well clearly this is a in-context operation and i have operation amnesia so this will be annoying but
if i moved forward
it
well
slender walrus
you've been able to apply these things
do you 100% agree that
(x+b/2)^2 = x^2 + bx + (b/2)^2
(you did the expansion yourself)
simple yes/no question
gritty hatch
x^2 15 div2
7.5.....pow2...
56.25?????
that already shattered damnit
semantics fail me again
slender walrus
what shattered?
if you're starting with
$$x^2 + 15x$$
then yes, $\br{\frac{15}{2}}^2 = 7.5^2 = 56.25$\
is the value you want to use to complete the square
grand pondBOT
ℝαμOmeganato5
last arch
(x^2+15x+7.5^2)=(x+7.5)^2
Sub any x into a calculator and you should see both are equal
slender walrus
i'm still not seeing what your issue is
slender walrus
you've literally just demonstrated you could do it
gritty hatch
its like trying to build a sand castle in the sea
but every time you try the water will wash it away and my memories will be erased
slender walrus
are you ignoring this question?
do you 100% agree that
(x+b/2)^2 = x^2 + bx + (b/2)^2
slender walrus
i've asked that like 10 times now
slender walrus
because that's the basic principle behind this
that's what you'll be applying every time for these completing the square questions
gritty hatch
i literally do not care
at all
like that has absolutely nothing to do with anything i asked
that doesnt even exist
you just presupposed it did
dude
slender walrus
wdym presupposed
gritty hatch
if (b/2)^2 is akin to the 5 in 5x when you need to create a term to make a perfect square so you do (5/2)^2
you do see how that isnt real right
you do that but that wont work
slender walrus
wdym won't work
gritty hatch
it wont work for all numbers and prove that dividing a number by 2 and squaring it gives a perfect square in the first place!
prove that
thats what i was asking the whole time
slender walrus
wdym
gritty hatch
literally nothing else mattered
NOTHING
at all
nothing
it was all meaningless
slender walrus
it works with all numbers
gritty hatch
it wont work for all numbers and prove that dividing a number by 2 and squaring ...
all that mattered was this
slender walrus
can you show me an example where it doesn't work?
slender walrus
(x + b/2)^2 expands to x^2 + bx + (b/2)^2
show me where that fails
slender walrus
because of that expansion i told you to do and keep referring back to
the equality goes in both directions
(x + b/2)^2 expands to and is equal to x^2 + bx + (b/2)^2
means that if you have x^2 + bx + (b/2)^2
you can get back (x + b/2)^2
if you start with
x^2 + 5x,
your b is the coefficient of x, which is 5
and (b/2)^2 will be (5/2)^2
which means that x^2 + 5x + (5/2)^2
is a perfect square, (with b = 5)
which from what we set up is (x + 5/2)^2
slender walrus
all that mattered was this
that's literally what the thing i told you to expand says
$$\violet{\br{x+\frac b2}^2} = \red{x^2 + bx} + \blue{\br{\frac b2}^2}$$
the left side, clearly a perfect square \
expands out to the red part plus the blue part which is the square of half of the coefficient of $x$
grand pondBOT
ℝαμOmeganato5
slender walrus
thus starting with the red, and adding the blue results in a perfect square
you were able to expand, so you know
$$\violet{\br{x+\frac b2}^2} = \red{x^2 + bx} + \blue{\br{\frac b2}^2}$$
i.e.
$$\red{x^2 + bx} + \blue{\br{\frac b2}^2} = \underbrace{\violet{\br{x+\frac b2}^2}}{\text{perfect square}}$$
so if you had just the red part, just refer to what you know above to identify the value you want
$$\red{x^2 + bx} +\underline{\ \ \ \ \ } = \underbrace{\violet{\br{x+\frac b2}^2}}{\text{perfect square}}$$
grand pondBOT
ℝαμOmeganato5
slender walrus
this is the last time I'm going to try
midnight plankBOT
@gritty hatch Has your question been resolved?
tepid vale
How do I solve the bottom halve
I did the circled ones already?
midnight plankBOT
How do I solve the bottom halve
I did the circled ones already?
@tepid vale Has your question been resolved?
sleek cloud
Which part your struggling
tepid vale
Which part your struggling
15-26
midnight plankBOT
midnight plankBOT
Available help channel!
twilit field
Trying to understand this method for converting $101$ from binary to base 10
grand pondBOT
What a wonderful world!
twilit field
$101 = 1 \cdot 2^0 + 0 \cdot 2^1 + 1 \cdot 2^2 = 1+0+4=5$
grand pondBOT
What a wonderful world!
twilit field
So to me it seems like we're adding the place values
which is fine
but why $2^{n-1}, why not 10^{n-1}$
grand pondBOT
What a wonderful world!
nova yoke
because 101 is given as a binary number, not decimal
fading ore
but why $2^{n-1}, why not 10^{n-1}$
Each place represents a power of 2
It's the same reason a number like $114 = 1 \cdot 10^2 + 1 \cdot 10^1 + 4 \cdot 10^0$ in base 10
grand pondBOT
hiidostuff
fading ore
In fact, the number being expanded out like that is what it means for a number to be in base 10
twilit field
That does makes sense
fading ore
Binary is particularly awesome at what it does because there's only two symbols needed to represent each possible state of a digit
Computers can handle that bc its easy to differentiate nothing vs something
Rather than 3 vs 5
midnight plankBOT
midnight plankBOT
Available help channel!
viral dagger
is there any tricks to power modulo if the base and the modulo share a factor
viral dagger
like for example, i have $45^{2025}\mod 50$
grand pondBOT
skissue.in.a.teacup
viral dagger
45 and 50 share a factor of 5
any trickeries i can do for it to simplify it without having to do like the (-5)^2025 ect
heady plume
remainder (45)^2025/50 = remainder (9*5)^2025/50 = remainder 5^2023 * 9^2025 / 2
does this not work 
heady plume
Does ✅
surreal moon
**skissue.in.a.teacup**
Not that I know of, but this particular example can be done very easily with another trick
viral dagger
enlighten me 
surreal moon
$45\equiv -5$, $(-5)^2=25$. If $(-5)^{2k}\equiv 25$, then what is $(-5)^{2k+2}$ mod $50$?
grand pondBOT
SWR
viral dagger
25?
surreal moon
viral dagger
is finding the fact that 5^2k=25 mod 50 just pattern finding?
surreal moon
I just happen to know that 25²=625. Once I saw the repeat, it was obvious
midnight plankBOT
heady plume
**skissue.in.a.teacup**
umm, ow it'd be helpful to note: say gcd(a, n) = d, a^k mod n = gcd(a^k, n) [shouldn't generalize but works in this specific case]
For example here, 5^2025 = 50x + y forces y to be 25
and then you're only left to find 9^2025 (mod 50) which is doable
midnight plankBOT
Available help channel!
viral dagger
midnight plankBOT
viral dagger
.close nvm
midnight plankBOT
midnight plankBOT
Available help channel!
vagrant cairn
hii, can anyone guide me on this. I'm not really sure about my way of doing it TvT
midnight plankBOT
hii, can anyone guide me on this. I'm not really sure about my way of doing it T...
vagrant cairn
ohh right yea yeaa
rich mirage
Why do i feel sarcasm from this? 💀
vagrant cairn
HELP 😭😭
rich mirage
If y = 2x and y = 4
Can you write it as 2x=4?
Since essentially you are saying y=y
vagrant cairn
hm mhmm yupp
rich mirage
Its the same thing here
kindred wraith
wait when I try to factorise the equation, it's kinda weird like
It prolly caus that not how you factorise such equations
Look closely at the signs inside and outside the bracket
And look at the signs on the step before
slender walrus
missing () and incorrect expansion
12x^2 - 4x^2 isn't 10x^2
ideally you wouldn't expand like that if the goal is to factorise the entire thing
note that both terms have a common factor of (2x+3)
so its better to start with subtracting 3(2x+3) from both sides and factoring that out
kindred wraith
so its better to start with subtracting `3(2x+3)` from both sides and factoring ...
A would that help?
Caus all you did is transpose the term
kindred wraith
Wait which step you talking about
slender walrus
restart from the very beginning
kindred wraith
Ok wait a sec
I only looked at the mistake on the last step
slender walrus
mistakes happened before the last step
kindred wraith
mistakes happened before the last step
Yep jus noticed
But instead of subtraction shouldn't it be division
slender walrus
not ideal
you'd need to be more careful about how you write it
which is very annoying
so better to subtract, then factor
kindred wraith
so better to subtract, then factor
If you divide then do multiple it would be faster?
slender walrus
not really no
vagrant cairn
ok guys, I'm just gonnna start again 😹😹
alright, this better be it if not I'm gonna crash out 😭😭🙏
vagrant cairn
due to the way you divided by (2x+3)
wait so is my way of factorising wrong in the latest working?
slender walrus
its not quite correct due to the way you cancelled (2x+3)
and disregarded the solution where (2x+3) = 0
vagrant cairn
its not quite correct due to the way you cancelled (2x+3)
and disregarded the so...
oo okok...
sorry if this sounds like a stupid question but I really suck at this but like when I write it is there supposed to be 3 brackets then?
slender walrus
wdym
vagrant cairn
like when I write it, sorry I still don't 100% get the factorising (2x+3) part 😭😭
slender walrus
so its better to start with subtracting 3(2x+3) from both sides
vagrant cairn
ohhh so is it like you subtract it from both sides ald and then you're left with (2x+3)(x-1)=0 like that ?? 🤔🤔
vagrant cairn
no, it's just like my way of understanding it cus I really get it 😭😭
slender walrus
so its better to start with subtracting 3(2x+3) from both sides
midnight plankBOT
midnight plankBOT
Available help channel!
chilly cobalt
From the image, let ABCD and PQRS be trapezoids, circles O1 and O2 have the same radius.Suppose AB = 28, and CD = 46. Find the area of PQRS
midnight plankBOT
From the image, let ABCD and PQRS be trapezoids, circles O1 and O2 have the same...
mighty plaza
chilly cobalt
create MN perpendicular to SR and passes through B, and RT perpendicular to CD
then triangles NBQ , MBR, CTR are congruent
then notice RS = CD - 2CT, PQ = AB + 2CT implies RS + PQ =AB + CD = 74
no clue what to do to find MN though
anyone?
<@&286206848099549185>
final girder
wait I try
chilly cobalt
AB + 2(TR + CT) = CD
chilly cobalt
PQ+SR=12+12+x+x+y+y=2x+2y+24, where x=MR and y=NQ no? and MN=x+y
oh
im tweaking how did i not see that😭
viral dagger
are all your questions geo or can you solve everything else 
chilly cobalt
so ans should be (74)(18)/4
chilly cobalt
are all your questions geo or can you solve everything else <:hmmcat:83150880164...
yes😭
i am absoloutely horrible at geo man
viral dagger
yes is not a god damn answer
viral dagger
i am absoloutely horrible at geo man
TRUEEA
chilly cobalt
yes is not a god damn answer
i have algebra and geometry and theyre the same difficulty (probably?) i can kinda do the algebra ones but geo just sucks
final girder
why is it so complicated
viral dagger
i have algebra and geometry and theyre the same difficulty (probably?) i can kin...
do you have number theory/combi&prob?
chilly cobalt
why is it so complicated
its geo💔
viral dagger
geo hateclub
final girder
I love geometry but why is that so hard 😭
viral dagger
I love geometry but why is that so hard 😭
has bro seen oly geo 😭
chilly cobalt
im alright at num theory
chilly cobalt
I love geometry but why is that so hard 😭
oh boy..
chilly cobalt
so ans should be (74)(18)/4
=333 whoo yay
final girder
what grade is that geo
viral dagger
im alright at num theory
you ever competed for olys where you have to give steps?
nova pike
geo hateclub
learn to appreciate geo. like oly geo is soo fun
viral dagger
what grade is that geo
i think 8 oly?
chilly cobalt
what grade is that geo
i dunno man
final girder
idk what ply is
viral dagger
olympiad
final girder
oly
chilly cobalt
you ever competed for olys where you have to give steps?
ive never qualified😭
final girder
yea I didng learn that
chilly cobalt
ill try this year though
heady plume
T_T Geo haters pls check #competition-math , there's a Geo question sitting
viral dagger
which
heady plume
Prove that AD perp EP ._> Havta scrooooll
chilly cobalt
ive never done geo proofs before, how do you write them?
viral dagger
ive never done geo proofs before, how do you write them?
you skip them
viral dagger
Prove that AD perp EP ._> Havta scrooooll
ok that one
nova pike
ive never done geo proofs before, how do you write them?
actually i recommend egmo for you :)))
chilly cobalt
actually i recommend egmo for you :)))
i read it and my brain fried
heady plume
Just noticed there's a soln to it =_+
final girder
i read it and my brain fried
what grade u in
chilly cobalt
8th
midnight plankBOT
viral dagger
that hard at 8
olympiads i tell you :(
final girder
olympiads i tell you :(
olym0iads at 8 is crazy
chilly cobalt
i dont think these are from olympiads tho
viral dagger
olym0iads at 8 is crazy
i started at 5 🤷
chilly cobalt
real
viral dagger
damn arya
midnight plankBOT
Available help channel!
steady trail
but if I find eigenvalues that are not distinct from each other, can I still write the diagonal matrix?
midnight plankBOT
but if I find eigenvalues that are not distinct from each other, can I still w...
steady trail
Yes
robust isle
sometimes you can, sometimes you can't
like the identity matrix is diagonalizable, it's diagonal itself
but all the eigenvalues are 1, they're all equal to each other
steady trail
exotic pelican
the dimension of the corresponding eigenspaces need to match the algebraic multiplicity
steady trail
I found $\lambda_{1}=0,\lambda_{2}=6,\lambda_{3}=6$
grand pondBOT
Task Bot
exotic pelican
so 6 has an algebraic multiplicity of 2
steady trail
Yes
exotic pelican
find the corresponding eigenspace
robust isle
well here your original matrix is symmetric, if it doesn't work you screwed up
steady trail
Wdym?
exotic pelican
or two linearly independent eigenvectors
steady trail
But in this case can I write the diagonal matrix?
I think not
For now
I have to see if it is diagonalizable to write it, right?
(my case )
robust isle
Wdym?
it's called the spectral theorem, you can always diagonalize symmetric matrices
if you haven't seen it forget it or not
steady trail
I didn't do it
robust isle
I have to see if it is diagonalizable to write it, right?
yeah fine check the eigenspaces then
steady trail
Ok I'll show my work in 5 minutes
Computing...
I think it's diagonalizable
Because it turns out that the algebraic and geometric multiplicity for lambda=1 they coincide
strong stump
it's called the spectral theorem, you can always diagonalize symmetric matrices
@robust isle so its diagonalizable for that
steady trail
The rank of A' i.e. where I put lamda=6 seems to me to be 1 but I'm not sure
robust isle
yeah it's rank 1
steady trail
However, if it is 1 it is diagonalizable
strong stump
i.e for that theorem its obvious that that the geometric multiplicity coincided with the arithmetic one, yes ?
@robust isle
robust isle
yeah
but let's not use it if task bot hasn't heard of it
it doesn't give you the diagonalization anyway
steady trail
yeah it's rank 1
How do I know for sure that the rank is 1, i.e. should I consider all minors of order 2 and make them the determinant?
strong stump
yes i mean he can use this to prove that he did well
robust isle
How do I know for sure that the rank is 1, i.e. should I consider all minors of ...
I just eyeballed it
all cols are multiples of (-1 2 -1) right
done
steady trail
,rotate
grand pondBOT
steady trail
This is the diagonal matrix, right?
robust isle
you can pick this one yes
steady trail
For lambda =0 , i found eingevector = (t,-2t,t) which I can write by setting t=1 as (1,-2,1)
steady trail
,rotate
grand pondBOT
robust isle
what's {(1, 1, 1)} supposed to be, a basis of eigenvectors for lambda=6 ?
you have a 2-dimensional set of solutions, you need to pick 2 linearly independent eigenvectors
steady trail
So I can't use the same ones?
robust isle
no
midnight plankBOT
midnight plankBOT
Available help channel!
snow pumice
midnight plankBOT
gusty falcon
can you use taylor expansion?
snow pumice
Don't know those yet :/
gusty falcon
what do you know?
snow pumice
A sht ton of theorems which are useless here - l' Hospital, sandwich, Bolzano-Weirestrass. Not a single practical one for this
I saw somebody rewrite the denominator as cos(pi(1/x)) and then do some voodoo, but idk what it was
gusty falcon
i mean it’s 0/0 so
atomic magnet
snow pumice
How am I so dumb that I thought the argument of cos was going to inf? wtf
thanks, my finals finna be cooked
gusty falcon
lol
snow pumice
.close
midnight plankBOT
midnight plankBOT
Available help channel!
graceful urchin
32x = 7 mod 45
I got to the point where 1 = 5 * 45 - 7 * 32 using euclidean algorithm
midnight plankBOT
32x = 7 mod 45
I got to the point where 1 = 5 * 45 - 7 * 32 using euclidean alg...
graceful urchin
and I am stuck here
heady plume
32x = 7 mod 45
I got to the point where 1 = 5 * 45 - 7 * 32 using euclidean alg...
multiply equation by 7
graceful urchin
7 = 35 * 45 - 49 * 32
heady plume
plug this into the modular equation and proceed
graceful urchin
I honestly don't know how 😁
graceful urchin
ok
heady plume
x = -4 mod 45
graceful urchin
ok
final asnwer is -4?
but how did we get from 32x = - 49 * 32 mod 45
to
x = - 49 mod 45
heady plume
but how did we get from 32x = - 49 * 32 mod 45
to
x = - 49 mod 45
dividing by 32 on both sides
graceful urchin
and now we need to add 45
so we will get positive integer
so final asnwer has to be 41
heady plume
but how did we get from 32x = - 49 * 32 mod 45
to
x = - 49 mod 45
jk we multiply with (32)^{-1} on both sides to get rid of 32, and are left with x = -49 = -4 = 41 (mod 45)
graceful urchin
I've gotta solve lots of this
heady plume
,calc (32*41 - 7)/45
grand pondBOT
Result:
29
an integer mwahaha
graceful urchin
what u think about this one
27x = 12 mod 40
I did it using ur method
and I got asnwer of 36
heady plume
,calc (27*36 - 12)
grand pondBOT
Result:
960
✅
heady plume
and I got asnwer of 36
36 is right
heady plume
.close
ripe bronze
What did I do wrong
midnight plankBOT
What did I do wrong
cunning ore
okk
ripe bronze
The video solved for A first though
So like is there a variable you gotta solve for first
cunning ore
lol
ripe bronze
https://youtu.be/0JQeo0yC9OQ?si=EZ1jWCs8Aoso0FS8
This is the video
7:49
For A i got 0.318
A is .75 in the video
@cunning ore
cunning ore
ah i think you ust wrote a numer wrong down
its -12.95
not -22.95
arae you allowed to use calculator?
cunning ore
yessir
ripe bronze
so i would just feel bad for you lol
So how do you do the multiplication or division with numbers with decimal.. like 39.01918 ×/÷78.0012... is there a secret method?
ripe bronze
Ok yes -12.95 got me the correct answer ty
Im always making some sort of mistake with something
tidal turret
midnight plankBOT
static maple
odd solar
just let me row reduce Meb
isn't M_EB the matrix that transforms standard basis to B?
tidal turret
yes
the columns of Meb(f) mean
(f(e1))_B = (1,1,1) and so on
,w det {{1,-2,a},{a,-5,1+a},{1,a,0}} = 0
grand pondBOT
tidal turret
,w rank {{1,1,-2,1},{1,1,-5,2},{1,1,1,0}}
grand pondBOT
tidal turret
$\polylongdiv{x^3-x^2+2x-2}{x-1}$
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
tidal turret
,w roots x^2 + 2
grand pondBOT
tidal turret
I see
we can use det = 0 to find a
and we can use Rational Root Theorem for finding real roots like a = 1
then after we found a = 1 is a root for the polynomial
we perform long division or synthetic division
we reduced the poly to a quadratic
then we notice we only have imaginary roots
so a = 1 is the only real a
Now the exercise is reduced to finding a basis for Im(f)
,w rref {{1,1,-2,1},{1,1,-5,2},{1,1,1,0}}
grand pondBOT
tidal turret
,w {{1,0,1},{-1,1,0},{1,-2,0}} * {{1,1,-2,1},{1,1,-5,2},{1,1,1,0}}
grand pondBOT
tidal turret
Im(f) = <(2,0,-1),(-1,-3,8)>
midnight plankBOT
@tidal turret Has your question been resolved?
odd solar
@tidal turret im not really understanding what you did to get a?
tidal turret
if you notice M_EB(f) is 4x3
but first column
is not dependant on a
also recall
det(A) = det(A^T)
if you take the determinant = 0 of last 3 columns
we get a cubic polynomial
odd solar
sure
tidal turret
<@1187916384341078070> im not really understanding what you did to get a?
is there some part that you are not understanding?
we can only take determinant of square matrices
odd solar
what does the det of the last 3 columns tell you about f though?
tidal turret
like
M_EB(f) matrix represents
(f(e1))_B = (1,1,1)
(f(e2))_B = (1,a,1)
(f(e3))_B = (-2,-5,a)
(f(e4))_B = (a, 1+a, 0)
now, the image of f is determinted by the outputs of f(e1), f(e2), f(e3), f(e4) but with respect to the canonical basis
tidal turret
what does the det of the last 3 columns tell you about f though?
it tells me we have some linear dependency in the image, specially for some values of a
det(A) = det(A^T)
for a = 1
we have some columns that are linearly dependent
Rank = dim(Im) = dim(column space) = dim(Row Space)
what is the column space? the span of the columns of the matrix
what is a span? the set of all possible combinations of the columns
we are trying to find for which a, does the kernel have a dimension of 2
kernel gets larger when we have linearly dependent columns or rows
specifically we need two linearly independent columns and two linearly dependent rows
odd solar
hmm okay, I think I follow. I'll leave to somebody to check if it's right though
tidal turret
like, the step of counting the pivots in M_EB(f) was not even necessary
,w {{1,0,1},{-1,1,0},{1,-2,0}} * {{1,1,-2,1},{1,1,-5,2},{1,1,1,0}}
grand pondBOT
tidal turret
,w rref {{2,2,-1,1},{0,0,-3,1},{-1,-1,8,-3}}
grand pondBOT
tidal turret
if we rref M_EE(f) after plugging a = 1 we still get first and third columns as pivots
I am pretty sure im right tbh
but I dont have the answer sheet
so idk
tidal turret
,w {{1,0,1},{-1,1,0},{1,-2,0}} * {{1,1,-2,a},{1,a,-5,1+a},{1,1,a,0}}
grand pondBOT
tidal turret
,w rref {{2,2, a-2, a},{0,a-1,-3,1},{-1,1-2a,8,a-2(a+1)}}
grand pondBOT
lucid rivet
.help
midnight plankBOT
Commands:
- clopen:
.close,.reopen - consensus:
.poll - factoids:
.tag - help:
.help - version:
.version
Type .help <command name> for more info on a command.
lucid rivet
Need help on this geometry
ocean meteor
do u know the formula of the sum of internal angles from a regular poligon?
S = (number of sides - 2)*180
midnight plankBOT
@tidal turret Has your question been resolved?
midnight plankBOT
@tidal turret Has your question been resolved?
midnight plankBOT
midnight plankBOT
Available help channel!
viral dagger
leg there be positive reals $x,y$, find the minimum of $\sqrt{4+y^2}+\sqrt{x^2+y^2-4x-4y+8}+\sqrt{x^2-8x+17}$
midnight plankBOT
leg there be positive reals $x,y$, find the minimum of $\sqrt{4+y^2}+\sqrt{x^2+y...
grand pondBOT
skissue.in.a.teacup
viral dagger
idk if this helps but $$\sqrt{y^2+4}+\sqrt{(x-2)^2+(y-2)^2}+\sqrt{(x-4)^2+1}$$
grand pondBOT
skissue.in.a.teacup
zealous schooner
What have you tried?
viral dagger
**skissue.in.a.teacup**
basically this
my intuition says x=y=2 but i have no idea of thats true or not
zealous schooner
Many times questions like these have geometric answers
Where each of the square root term acts like the hypotenuse of a certain right triangle
viral dagger
whar
zealous schooner
The sqrt(y^2+4) term is the hypotenuse of a right triangle with sides y and 2
viral dagger
oh
how can i use that to find the minimum tho
is it like if you combine them they make some sort of straight line or smth?
zealous schooner
is it like if you combine them they make some sort of straight line or smth?
Exactly
I'm not sure how to put this idea into words
viral dagger
wait but wouldnt that be like the maximum
zealous schooner
Ignore the labeling, but this kind of diagram is what I want you to imagine
Then OA + AB + BC >= OC with equality precisely when all the slopes are equal
viral dagger
ohhh
olive matrix
Ignore the labeling, but this kind of diagram is what I want you to imagine
neat
viral dagger
for some reason i switched them up being that the whole sum is the orange line instead
zealous schooner
for some reason i switched them up being that the whole sum is the orange line i...
That's what you want for the minimum
Note that this is only useful if the orange line has constant length
viral dagger
but its not constant
viral dagger
Ignore the labeling, but this kind of diagram is what I want you to imagine
btw what did you use to make this?
zealous schooner
btw what did you use to make this?
I found it on quora lol
zealous schooner
<:hmmcat:831508801646755850> but its not constant
It is
If you choose the signs correctly
viral dagger
signs?
zealous schooner
What did you do?
viral dagger
wdym
zealous schooner
Why do you think it isn't constant
viral dagger
im not sure qhat you mean with choosing signs
zealous schooner
Remember that the sides of a triangle have to be positive
viral dagger
i mean sqrt((2x-5)^2+(2y)^2) isnt constant is it?
zealous schooner
i mean sqrt((2x-5)^2+(2y)^2) isnt constant is it?
I told you not to focus on the labeling
viral dagger
Remember that the sides of a triangle have to be positive
so x>=5 and y>=2
zealous schooner
so x>=5 and y>=2
No
sqrt((x-2)^2+(y-2)^2) is the hypotenuse of a right triangle with sides |x-2| and |y-2|
viral dagger
oh im dumb
wait so can you write the base as (2) and (x-2) and (4-x) and the height as (y) and (2-y) and (1)
ohh so orange is 5?
zealous schooner
wait so can you write the base as (2) and (x-2) and (4-x) and the height as (y) ...
Only if 2 <= x <= 4 and 0 <= y <= 2
viral dagger
ok so if you let them have the same slope there exisfs a sol for x and y
zealous schooner
So find the solution
viral dagger
Only if 2 <= x <= 4 and 0 <= y <= 2
and for x,y outside of tthe bounds the orange line is bigger than 5 right?
zealous schooner
and for x,y outside of tthe bounds the orange line is bigger than 5 right?
Why?
viral dagger
So find the solution
8/3 and 3/2
viral dagger
brb sorry
x>4 would mean sqrt(y^2+4)+sqrt((x-2)^2+(y-2)^2)+sqrt((x-4)^2+1)>sqrt(4)+sqrt((x-2)^2)+sqrt((x-4)^2+1)=x+sqrt((x-4)^2+1)>x+sqrt(1)>4+1=5
heady plume
8/3 and 3/2
Umm, is this okay? (2, 1) giving me f(2,1) = 2√5 + 1 < f(8/3, 3/2)
zealous schooner
Umm, is this okay? (2, 1) giving me f(2,1) = 2√5 + 1 < f(8/3, 3/2)
We're trying to find the minimum
It's greater
heady plume
What's f(8/3, 3/2)?
zealous schooner
x>4 would mean sqrt(y^2+4)+sqrt((x-2)^2+(y-2)^2)+sqrt((x-4)^2+1)>sqrt(4)+sqrt((x...
Sure
But you don't need to do all this
Just use the triangle inequality
heady plume
Yeah nope, I'm still getting f(8/3, 3/2) > f(2, 1)
viral dagger
Just use the triangle inequality
wait what
zealous schooner
wait what
The method I sent above with the triangles is just the triangle inequality in R^2
viral dagger
Yeah nope, I'm still getting f(8/3, 3/2) > f(2, 1)
zealous schooner
$\sqrt{x_1^2+y_1^2}+\dots +\sqrt{x_n^2+y_n^2}\ge \sqrt{(x_1+\dots +x_n)^2+(y_1+\dots +y_n)^2}$
grand pondBOT
kheerii
quiet hinge
Ignore the labeling, but this kind of diagram is what I want you to imagine
this thing
zealous schooner
the image tells you everything you want here
(Like when equality holds)
viral dagger
if you apply that directly, the sum is geq to sqrt((2+x-2+x-4)^2+(y+y-2+1)^2)=sqrt((2x-4)^2+(2y-1)^2)
ehh i dont think this works
zealous schooner
if you apply that directly, the sum is geq to sqrt((2+x-2+x-4)^2+(y+y-2+1)^2)=sq...
Consider only positive values
It makes the inequality stronger
viral dagger
wdym only positive values
viral dagger
**skissue.in.a.teacup**
the only negatives here are incorporated into the squares tho?
oh wuat
zealous schooner
$\sqrt{x_1^2+y_1^2}+\dots +\sqrt{x_n^2+y_n^2}\ge \sqrt{(|x_1|+\dots +|x_n|)^2+(|y_1|+\dots +|y_n|)^2}$
grand pondBOT
kheerii
zealous schooner
This is what I mean
To get the strongest inequality you want the right side to be as large as possible, which happens when you take all the terms to be positive
viral dagger
for x<2, sqrt((2+2-x+4-x)^2+(2y-1)^2)=sqrt((8-2x)^2+(2y-1)^2)
like this?
for the y it wiuld be the same?
zealous schooner
for x<2, sqrt((2+2-x+4-x)^2+(2y-1)^2)=sqrt((8-2x)^2+(2y-1)^2)
That does not look right
zealous schooner
You're complicating things a bit
Just use the triangle inequality with the absolute values and see what you get
viral dagger
triangle ineq us tge |x+y|<=|x|+|y| right
midnight plankBOT
@viral dagger Has your question been resolved?
viral dagger
i cant think of how triangle ineq can be used here
heady plume
viral dagger
whar
heady plume
i cant think of how triangle ineq can be used here
You are to minimize PQ + PR + PS such that PI = 2, PJ = 1. Play around with P and see why what you're doing
viral dagger
wtf
heady plume
For starters, it's immediately evident that P must lie within the ∆QRS and to minimize the distance to vertices, must be their centroid
Reference to Fermat Point of a triangle
viral dagger
what is this bro 😭
heady plume
P(x, y), Q(4, y - 1), R(2,2), S(x - 2, 0) is what it is
PQ + PR + PS is given expression
viral dagger
find real $x>0$ such that $$\log_2(x)\log_4(x)\log_6(x)=\log_2(x)\log_4(x)+\log_2(x)\log_6(x)+\log_4(x)\log_6(x)$$
midnight plankBOT
find real $x>0$ such that $$\log_2(x)\log_4(x)\log_6(x)=\log_2(x)\log_4(x)+\log_...
grand pondBOT
skissue.in.a.teacup
viral dagger
modern sapphire
take the common base and simplify ig
modern sapphire
$log_a(x) = \frac{log_{10}(x)}{log_{10}(a)}$
grand pondBOT
Facter10Br4g
heady plume
**skissue.in.a.teacup**
this is basically, $\log_x 2 + \log_x 4 + \log_x 6 = 1$
grand pondBOT
Arya
heady plume
assuming x \neq 1
carmine sigil
midnight plankBOT
@viral dagger Has your question been resolved?
olive yew
**skissue.in.a.teacup**
Can we have ((logx)/(log2))*(logx)/(log4)
(Log2(x))(log4(x)) = ((logx)^2)/((log2)(log4))
viral dagger
wa sorry
heady plume
viral dagger
how 😭
heady plume
Yes lol, $\log_a b = \frac{1}{\log_b a}$
fallen galleon
so on and so forth
grand pondBOT
Arya
viral dagger
lemme see
fallen galleon
you'll end up with a lot of fractions but clearing them gives you that
viral dagger
X = 48 <@750992393502720010>
!nosols :p
midnight plankBOT
X = 48 <@750992393502720010>
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
olive yew
!nosols :p
Alr
heady plume
X = 48 <@750992393502720010>
and no, this is wrong incomplete btw
olive yew
and no, this is ~~wrong~~ incomplete btw
Is it ?
olive yew
and no, this is ~~wrong~~ incomplete btw
Let me write the solution
viral dagger
ok so this
viral dagger
right
heady plume
$\implies 1 = \frac{1}{\log_2 x} + \frac{1}{\log_4 x} + \frac{1}{\log_6 x}$
grand pondBOT
Arya
olive yew
((Log(x))^2)/(log2)(log4)+((Log(x))^2)/(log2)(log6)+((Log(x))^2)/(log4)(log6)
viral dagger
ohh
heady plume
uff
viral dagger
ok ic
olive yew
((Log(x))^2)/(log2)(log4)+((Log(x))^2)/(log2)(log6)+((Log(x))^2)/(log4)(log6)
Now multiply this with (log2)(log4)(log6)
viral dagger
so basically its 1/logx2logx4logx6=logx6/logx2logx4logx6+logx4/logx2logx4logx6+logx2/logx2logx4logx6 => 1=logx6+logx4+logx2=log_x 48
so x^1=48=>x=48
heady plume
Now multiply this with (log2)(log4)(log6)
and ?
heady plume
You got this @viral dagger ?
viral dagger
what?
viral dagger
so basically its 1/logx2logx4logx6=logx6/logx2logx4logx6+logx4/logx2logx4logx6+l...
i mean i did it like this
olive yew
I'm LHS we have (log(x))^3 = ((log(x))^2)(log2 + log4+ log6)
Simplifying (log2 + log4 + log6)
= log(48)
heady plume
Yes, this translates to $\log_x 2 + \log_x 4 + \log_x 6 = 1$
grand pondBOT
Arya
heady plume
so you're mostly done :p
heady plume
(Log(x))^3 = ((log(x))^2)(log48)
factorise =_=
@heady plume
midnight plankBOT
midnight plankBOT
Available help channel!
last slate
1.For every $\epsilon >0$,there are infinity many n such that the distance between $a_n$ and 0 is less than $\epsilon$.
midnight plankBOT
1.For every $\epsilon >0$,there are infinity many n such that the distance betwe...
grand pondBOT
last slate
Need to write the negation for this
This is what I wrote 1.For atleast one $\epsilon >0$,there are infinity many n such that the distance between $a_n$ and 0 is greater than $\epsilon$.
grand pondBOT
radiant roost
yeah i don't think that's right
last slate
nice
so
what prof said is "whats the minimal change that is needed in the statement which can contradict it"
so for that reason i didnt change it to finite ways
radiant roost
but what you wrote is not a negation of the sentence
so you'll have to do something different
last slate
but there exists atleast one such epilon which satifies this condition then the original statement would be false right?
because it says
"for every epsilon"
radiant roost
no
let the sequence be 0,1,0,1,0,1...
let epsilon be 1/2
there are infinitely many n such that the distance between a_n and 0 is greater than 1/2
radiant roost
so there exists at least one such epsilon that satisfies this condition
last slate
let the sequence be 0,1,0,1,0,1...
is this sequence?
becuase lt at n tends infinity is not defined right
radiant roost
right
last slate
so its not a sequence?
radiant roost
it is a sequence
last slate
i will come back after working on epsilon delta definations and on how to write negation of a statement
.close
midnight plankBOT
last slate
thanks
midnight plankBOT
Available help channel!
dense arch
Hello. Im super confused how does this expression simplify to that. I've been trying on my end and I couldn't get it
midnight plankBOT
Hello. Im super confused how does this expression simplify to that. I've been tr...
sleek cloud
what have yu tried
heady sentinel
multiplying the 1/(1-x)^2 by (1-x)/(1-x)
We take out negative 1 common out of the numerator and multiply it with the negative 1 on the outside.
tired swan
anyone help on q
dense arch
We take out negative 1 common out of the numerator and multiply it with the nega...
what?
so you multiply by x-1 / x-1?
sleek cloud
did you get (x+1) / (1-x)^3?
sleek cloud
x+1 = -(-1-x) thats what they did
heady sentinel
what?
Here, multiply 1/(1-x)² by (1-x)/(1-x).
And we get, (1-x)/(1-x)³.
Now add the two as they have the same denominator.
We get (2x + (1-x))/(1-x)³.
Simplifying, we get, (1+x)/(1-x)³.
dense arch
bruh
sleek cloud
seems unnecesarry
midnight plankBOT
midnight plankBOT
Available help channel!
flint jewel
What is the sum of all the multiples of 3 from 15 to 48?
midnight plankBOT
What is the sum of all the multiples of 3 from 15 to 48?
gusty falcon
u want to include 15 and 48?
flint jewel
yes
gusty falcon
what have u learnt in class related to this?
flint jewel
I believe arithmetic sequence should suffice
gusty falcon
yes an arithmetic series
flint jewel
ysa
gusty falcon
sure
flint jewel
n i believe is 16 by doing manual math
gusty falcon
maybe you're familiar with an equivalent formula which uses the first and last term
these are the same but yeah that one is more applicable
flint jewel
i see
gusty falcon
n i believe is 16 by doing manual math
how'd you get it with manual math?
flint jewel
how'd you get it with *manual math*?
48/15, then lcf 16/3
gusty falcon
lcf?
gusty falcon
idk what that does
also 48/15 simplifies to 16/5
but that's a matter for another day ig
,, a_n = a_1 + (n-1)d
grand pondBOT
mmmm7
flint jewel
yes
gusty falcon
if u didn't put a_n
then there's 2 unknowns and one equation
u can't uniquely solve it
so it still doesn't make sense how u got -4
but yeah
flint jewel
I got this
12+3n=an
3n=-12
gusty falcon
you assumed a_n = 0
gusty falcon
u can also just do 48/3 - 15/3 + 1 where 48/3 counts all the multiples of 3 between [0,48] and likewise for 15/3
the + 1 is to count back the case where 15 isn't included
odd solar
Sn=(n/2)(a1+(n-1)d)?
Probably easier to use this
gusty falcon
they're using that
odd solar
For the summand 3n, then just figure out the bounds which is real simple
gusty falcon
they were solving for n in the process
odd solar
You made it a much longer process than it should have been
cerulean oyster
yk, summing from 1 to 100
odd solar
[ \sum_{n=x}^y 3n ]
grand pondBOT
shsgd
gusty falcon
it's not always about solving it the fastest way
read the chat ahead
we're using what they know
cerulean oyster
like, yes, this is true, but that doesnt solve anything about the problem, is more like re-writing the problem
gusty falcon
also it's 2 lines anyway
your method is not any faster
$S_n = \frac n2 \qty(2a_1 + (n-1)d)$ is equivalently written as $S_n = \frac n2 \qty(a_1 + a_n)$
grand pondBOT
mmmm7
grand pondBOT
Result:
378
me dumb
odd solar
your method is not any faster
It’s the same method, just far easier to see why n = 12
odd solar
Fine, i’ll leave you to it
gusty falcon
**shsgd**
no i just don't see how you expressing it in sum notation changes anything about the question
they already have a closed form for the sum for an arithmetic series
heady plume
I claim: 3[(16)(17) - (5)(4)]/2 is the fastest solution
odd solar
no i just don't see how you expressing it in sum notation changes anything about...
It doesn’t change anything about the question. It’s just describing the question mathematically. Then using summand 3n means the summation indices will be contiguous so it’s clear that there are 12 multiples of 3 between 15 and 48
midnight plankBOT
@flint jewel Has your question been resolved?
flint jewel
n was 7
flint jewel
in the summation formula
so there were 4 choices
one of them was 378 which was the highest
flint jewel
yes
heady plume
and we have learnt the formula: sum of an arithmetic series is (n / 2)[ 2a + (n - 1) d ]
right?
flint jewel
Yes
heady plume
so we require the "n", yes?
flint jewel
yes
heady plume
which implies, "how many terms are there in this series"
so you use 48 = 15 + (n - 1)*d , right?
flint jewel
the original n is 12 but since we start at 15 and not 3 i just did
12-15/3
heady plume
where d is common difference = 3
flint jewel
so you use 48 = 15 + (n - 1)*d , right?
yes
heady plume
so, 3n - 3 = 48 - 15 = 33, yes?
flint jewel
yes ig that should work
heady plume
no buts.... we used our first term and determined how many terms ahead is the last term, and that is what our n should be
flint jewel
Oh
nova pike
What is the sum of all the multiples of 3 from 15 to 48?
i dont want to disrupt the conversation, however what you can do is notice 15 = 3 * 5 and 48 = 3 * 16. Therefore the ones in between like 3 * 6, 3 * 7, 3 * 8, ...., 3 * 14, 3 * 15 are also included in the sum. so we want to find the sum of the numbers 3 * 5 + 3 * 6 + ... + 3 * 15 + 3 * 16. Moving the 3 out, we get 3(5 + 6 + 7 + 8 + 9 + ... + 15 + 16). Then there is this special formula that you need which is 1 + 2 + 3 + ... + (n-1) + n = n(n+1)/2
flint jewel
i dont want to disrupt the conversation, however what you can do is notice 15 = ...
Oh wait that also make sense
heady plume
Yes, now all's left is to plug into formula: (n / 2)( a + l ) = 6*(15 + 48)
,calc 6*63
grand pondBOT
Result:
378
._.
nova pike
Oh wait that also make sense
ok, so you understand everything up to the part where our answer becomes 3(5 + 6 + 7 + ... + 14 + 15 + 16). (again sorry for disrupting the conversation lol)
heady plume
?
flint jewel
oH i used a different method
heady plume
(n / 2)( 2a + (n - 1)d) = (6)(2*15 + 11*3) = 6*(63)
flint jewel
i dont want to disrupt the conversation, however what you can do is notice 15 = ...
Are u a math competitor by anychance 👀
It seems i was blinded
flint jewel
yes
heady plume
why you plugging n = 16
nova pike
Are u a math competitor by anychance 👀
um yes, how do you know?
flint jewel
um yes, how do you know?
cuz im currently gonna participate for a math quiz bee and found that math competitors have like some type of different analysis from the normal peeps so i find it very intriguing
heady plume
bless your quiz bee
nova pike
cuz im currently gonna participate for a math quiz bee and found that math compe...
ah interesting. i mean i always just try to find the easiest way to be fair, but you can definitely use the arithmetic series formula for this problem, basically what you are trying to do now
flint jewel
ah interesting. i mean i always just try to find the easiest way to be fair, but...
ooo, how do u find the easiest ways
do u first find a pattern between each numbers?
heady plume
3(5 + ... + 16) = 3*(16-5+1)(16 + 5)/2
nova pike
do u first find a pattern between each numbers?
yeah, sometimes i guess the pattern and then proceed to prove it
heady plume
is another way to do it
heady plume
i dont want to disrupt the conversation, however what you can do is notice 15 = ...
this is what adinfinitium said basically
nova pike
In my opinion, the only thing you really need to use is 1+2+3+....+n = n(n+1)/2, then you just apply the formula carefully
nova pike
In my opinion, the only thing you really need to use is 1+2+3+....+n = n(n+1)/2,...
you can try prove this first if you like (theres a nice way of proving it). but otherwise notice that we want to compute say X = 3(5 + 6 + .... + 15 + 16). Notice (5 + 6 + ... + 15 + 16) is just the same as (1 + 2 + 3 + 4 + 5 + 6 + ... + 15 + 16) without the (1 + 2 + 3 + 4). Then you can use the formula I provided and youre done
last slate
how do i factorize the denominator?
midnight plankBOT
how do i factorize the denominator?
grim vector
Vertical asymptote is when denominator = 0
normal root
there is
grim vector
so there is no vertical asymtote?
Why so ,
last slate
well i solve these questions based off the examples they give me and im supposedly factoring out the gcf first
heady plume
so there is no vertical asymtote?
Yes, there's no vertical asymptote, just an isolated point discontinuity