help-27

[woven radish]

mercy

[potent compass]

Why divide it

[restive river]

To factor by 2

Suggestion is if x ^2 has coefficient a then factor x terms by a then do completing square in bracket

Else

$ax^2 + bx + c = a(x^2 + \frac{b}{a}x + (\frac{b}{2a})^2) + c - a(\frac{b}{2a})^2$\

$ax^2 + bx + c = a(x + \frac{b}{2a})^2 + c - \frac{b^2}{4a}$

[woven radish]

mercy

[restive river]

Like this

[potent compass]

Oh

I’m knok

Ok

Tysm

Omds my WiFi

Tysm everyone

.close

[devout snow]

Channel closed

Closed by @potent compass

Use .reopen if this was a mistake.

[potent compass]

.reopen

[devout snow]

✅ Original question: #help-27 message

[potent compass]

Could u pls help me again

[winged tapir]

what do u need help with

[potent compass]

Number 3

Idk if u can see well

My eyesit is bad

[winged tapir]

-x^2 + 6x + 9 is NOT equal to -(x+3)^2

[potent compass]

Oh

Do I factor out the negative first?

[winged tapir]

probably, it will make it easier to understand

[potent compass]

But then what do I do next

[winged tapir]

so you have -x^2 + 6x = -8 right

[potent compass]

Oh..

[winged tapir]

solve x^2 - 6x = 8

add 9 and subtract 9

[potent compass]

Yes sir/maam

[winged tapir]

lol

[potent compass]

Ty’sm

[winged tapir]

np

[potent compass]

I’ll say splitter longer in case I need help

Can someone help me w this

Will the sign change pls

[pliant tusk]

wdym sign change?

[potent compass]

The left fraction

• and -

[pliant tusk]

huh 😭

be more specific im lost

[winged tapir]

btw u factored it wrong

[potent compass]

Dang it

[pliant tusk]

you can take out the negative and make the entire thing addition

[winged tapir]

x^2 -4 is NOT equal to (x-2)(x-2)

[pliant tusk]

^

[potent compass]

There’s a negative tho

[potent compass]

Ty

[pliant tusk]

it is still difference of squares

a^2-b^2 = (a+b)(a-b)

oh i see where ur confused

you tried switching 4–x^2 to x^2-4

[potent compass]

Yes

[crude niche]

4-x^2 is not x^2-4

look at the signs

[pliant tusk]

ok well if you do that then it becomes + 2/(x^2-4)

[crude niche]

2 / -(x^2-4)*

[potent compass]

What if -(x-2)

[pliant tusk]

i took out the subtraction made it addition

[crude niche]

that is x+2

[crude niche]

ah

[potent compass]

Then what do I do

It’s confusing

[pliant tusk]

do this

does that make sense

[crude niche]

try making common denominators

[crude niche]

after this

[potent compass]

Don’t I have to reduce the x^2-4

[potent compass]

Yes

[pliant tusk]

yes do that now

[crude niche]

don’t

[crude niche]

why?

[pliant tusk]

she means factor

[potent compass]

Oki

[crude niche]

why factor though

i was thinking common denominators

[pliant tusk]

factor so left fraction just needs a x+2

its the same thing

[crude niche]

ah

[crude niche]

just more steps

[pliant tusk]

clearer at this level

[crude niche]

fair

[potent compass]

So I first change it but make it still be x^2 -4

Then I reduce it

[pliant tusk]

yes sure

[potent compass]

Tysm everyone

Nvm

.close

[devout snow]

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[devout snow]

Available help channel!

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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[elfin mist]

Any ideas on how to solve?

[trail eagle]

Can you compute b-a and c-a with log rules?

[elfin mist]

what do you mean whit compute?

[trail eagle]

Like simplify

[elfin mist]

Like this?

I feel like I fissed a formula

[trail eagle]

No

[elfin mist]

Like this?

[trail eagle]

,tex .log rules

[woven radish]

Azyrashacorki

[trail eagle]

That works

There's one last thing you can do with the numerator

Noticing that 1/3 = 3^{-1}

[elfin mist]

If I remmeber correctly radicals share some methods with logs

So the rule works like this right? Sorry for being dumb, it's kinda late

[trail eagle]

Yes that's exactly it

[elfin mist]

thank you

.close

[devout snow]

Channel closed

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[devout snow]

Available help channel!

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Remember:
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• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[potent compass]

Question 3

[azure ledge]

"Can someone please tag the bot or send a link for the 'Bartle Real Analysis' solutions PDF? I can't find it in the search

[potent compass]

This is occupied

[trail eagle]

Where did 36k come from?

[potent compass]

Oh

Mb

I’ll re do it

[trail eagle]

Ok

[potent compass]

Got it, thankie

[frank rivet]

Hello

[trail eagle]

Hi! If you need help, refer to #❓how-to-get-help to see how to claim your own channel.

[frank rivet]

Thx.

but I don't need to help. I can help you with maths.

[trail eagle]

That's good as well! You're free to contribute in currently open help channels.

[candid maple]

if you wish to help, you may look around in the forum and currently occupied channels to see if any still needs help that you can help with.

[trail eagle]

OP seems to have figured it out for this problem.

[potent compass]

Can I get help w this

For the graph ones

I understand the rules

But idk how they apply to the graphs

[trail eagle]

Well if the graph doesn't touch the x-axis it immediately has no real roots

Is that clear?

[potent compass]

Yes pls

[trail eagle]

If there is just one intersection, then it has to be at the vertex of the parabola, and that indicates that the discriminant is 0 (then you get +- 0 in the quadratic formula, so there's only one case).

[potent compass]

Idk this

[trail eagle]

In the quadratic formula $$x_{1,2} = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a},$$ the only way you can get 1 solution (so 1 intersection), is if $\Delta = 0$, in which case you get $$x_{1,2} = \frac{-b \pm \sqrt{0}}{2a} = \frac{-b}{2a}.$$ Notice how now the too roots $x_1$ and $x_2$ you would get are the same.

[woven radish]

Azyrashacorki

[trail eagle]

So there's only one solution, and thus one intersection

[potent compass]

What would it look like

[trail eagle]

Like graph 3 in the first row or graph 2 in the second row

[potent compass]

What would equal look like

[potent compass]

Ooo

How would Ik it’s rational and equal or unequal pls

[trail eagle]

If $\Delta = 0$, then you get above that the root is $x_1 = \frac{-b}{2a}$, which is rational whenever $\frac{-b}{2a}$ is rational.

[woven radish]

Azyrashacorki

[trail eagle]

So you can compute -b/2a and see if it is rational

[potent compass]

W a graph?

I still don’t get you

[trail eagle]

The root is at -b/2a in graphs c) and e)

That root is rational if you compute -b/2a and it's rational.

[potent compass]

Wait where did -b/2a come from

[trail eagle]

.

[potent compass]

Oh ok

But I added it myself

So how can I tell it without the -b/2a

[trail eagle]

That is how you tell

You can't tell just from the graph without information about whether -b/2a is rational or not

[potent compass]

Oh ok ok

Ty!

What about unequal and equal

[trail eagle]

Well we've just discussed the equal part.

If $\Delta = 0$, then there is a single root (so it looks like graphs c) or e)) and you tell whether the root is rational or not depending on whether $\frac{-b}{2a}$ is rational or not.

[woven radish]

Azyrashacorki

[potent compass]

I don’t get it

Ho w would delta be 0 on a graph

[trail eagle]

When there is only one root, like in graphs c) and e).

[potent compass]

Ohh ok

[trail eagle]

If the roots are not equal, that is $\Delta > 0$, then in order to get a rational root, you need $\Delta$ to be a perfect square, so that $\sqrt{\Delta}$ is an integer. In that case this automatically implies that the whole expression in the quadratic formula is rational, so your roots are rational. \

If $\Delta$ is not a perfect square, then the roots will be irrational.

[woven radish]

Azyrashacorki

[potent compass]

Oki oki

Tysm

Number 7

[trail eagle]

For unequal roots you need $\Delta > 0$.

[woven radish]

Azyrashacorki

[trail eagle]

Also $a = r$, not 1

[woven radish]

Azyrashacorki

[potent compass]

Oh

[potent compass]

But how do I solve it

[trail eagle]

That's good but as I said this needs to be a strict inequality to get unequal roots.

[potent compass]

Oh

How do I do tht

[trail eagle]

It's the same thing just get rid of the $\ge$ and replace them with $>$.

[woven radish]

Azyrashacorki

[potent compass]

Why Pls

[trail eagle]

To get unequal roots you need $\Delta > 0$.

[woven radish]

Azyrashacorki

[trail eagle]

Not $\Delta \ge 0$

[woven radish]

Azyrashacorki

[trail eagle]

Because if $\Delta = 0$ you get equal roots.

[woven radish]

Azyrashacorki

[potent compass]

Do I do this for all real and unequal

[trail eagle]

Yes

[potent compass]

Hi

Ty

[devout snow]

@potent compass Has your question been resolved?

Channel closed

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[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[frank rivet]

Hello,

[frank rivet]

I can help you

[green crypt]

what is bro tryna do

😭

hey the people who want help start these channels

[strange ruin]

tf is this bruh

im crine

[green crypt]

or occupy em

[queen nimbus]

Can you read #❓how-to-get-help to understand how this works? and please only try to help when you know what you are talking about and can communicate that clearly with the people asking for help.

.close

[devout snow]

Channel closed

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Remember:
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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[pure cedar]

33

there are infinite sets A for which this is true right?

[acoustic leaf]

why should there be?

[pure cedar]

A can be {15}

[acoustic leaf]

what is f(1) then?

[pure cedar]

A can be {22}

[pure cedar]

31

[acoustic leaf]

how can a function output a value not in its codomain?

[uncut crow]

A \subset N

[pure cedar]

its not possible then right?

[bleak dirge]

The range should be equal to the codomain

[pure cedar]

cuz inputs are reals

[uncut crow]

A \subset N in the question just makes no sense to me

[pure cedar]

so am i right with A being {15}

[bleak dirge]

It is a quadratic

[bleak dirge]

No

[acoustic leaf]

it tells you the domain, which is the positive real numbers

[pure cedar]

so there isnt any A for which this is true?

[bleak dirge]

A will be the set of "output" values of the function

[uncut crow]

A \subset N seems like a mistake in the question

[pure cedar]

but we just cannot choose any A such that fxn becomes onto

[acoustic leaf]

i think that A subset N is only meant to be true for the first part? that's the only way it would make sense

actually it doesn't work even for the first part nvm

[pure cedar]

it is cbse board exam

💔💔

[glossy dew]

wtf makes no sense

thank god i got set 1

[bleak dirge]

The minimum value of the quadratic to infinity will be A

[pure cedar]

damn,

[void fox]

lol not like exam boards never make mistakes

[uncut crow]

amen 0lante

[void fox]

also the english is so poor like

Justify.

[bleak dirge]

I had set 2

[pure cedar]

but if we take A as {15}
isnt range = codomain simply

[void fox]

it's not even a complete sentence

[uncut crow]

find set A

[pure cedar]

IKR

[void fox]

amen layla

[pure cedar]

im gonna loose marks cuz of this and i hate it cuz everything else i have done correct

[bleak dirge]

No because the domain is R+

[glossy dew]

probably hurried since id be pissed to elaborate if i had to type 15 pdfs

[pure cedar]

so A does not exist right?

[bleak dirge]

😭😭

[pure cedar]

ill piss at cbse , if their incompetence does ts

[pure cedar]

am i right?

i thought of going modular arithmetic on ts

and i tried to find the largest A

such that every integer is possible

but since inputs are real

[bleak dirge]

A should be 15 to infinity

[worldly flame]

does it not mean the codomain is a subset of N

obviously wonky notation but idk

[pure cedar]

DOES NOT MAKE SENSE DUDE

[glossy dew]

no it was necessary for part A

[bleak dirge]

Cbse 💔

[glossy dew]

but omitted for part B

[uncut crow]

maybe just ignore \subset N and then it's a sensible question. who knows if it was the intended question though

[glossy dew]

wtf

[pure cedar]

no , A is the codomain, which is subset of N

[pure cedar]

I HATE TS

[worldly flame]

this reads like an indian exam

oh it is

[void fox]

That would be \supset, not \subset

[worldly flame]

subset of N*

[pure cedar]

whats supset

,, \supset

[woven radish]

parthisjoking

[uncut crow]

it makes no sense for part A. f(pi) for example, is an expression that is supposed to parse, since the domain of f is R_+

[pure cedar]

oh

this

[uncut crow]

but it does not if the codomain is a subset of N

[pure cedar]

what should i do now 💔💔💔

am i gonna loose these 5 marks

[uncut crow]

i have no idea

[pure cedar]

ok
come to the last part of the question

[worldly flame]

oh is N supposed to be the natural numbers

[pure cedar]

where we have to prove that f(a) > 7

[glossy dew]

yeah but blind cbse method gives\
$4x_1^2 + 12x_1 + 15 = 4x_2^2 + 12x_2 + 15 \implies 4(x_1-x_2)(x_1+x_2+3) = 0$\
and range of natural numbers is to invalidate the second possible equality

[void fox]

To find a set such that reducing the codomain to that set makes the function surjective is to essentially find the range of that function

[glossy dew]

yeah no omitted it is

[pure cedar]

the range is R+ tho

[void fox]

R+ is the domain

[pure cedar]

but in a quadratic

range is also

R+

nvm

[void fox]

what?

[pure cedar]

its range is 15 to infty

[void fox]

yes

[uncut crow]

isn't the last part quite simple if x >= 0 then 4x^2 + 12x + 15 >= 15 > 7

[glossy dew]

cbse is quite simple

[pure cedar]

yea but cbse doesnt give marks for that

you have to show it with derivatives

[uncut crow]

what the blud

you can't use that a sum of nonnegative numbers is nonnegative?

[pure cedar]

AND HERE THE FUNCTION IS NOT EVEN DIFFERENTIABLE

[pure cedar]

not really 🥰

we have to show them that we know a lot

[uncut crow]

lol

[pure cedar]

also

whenever we go for the exam

so we have to explain every answer like a baby

[rare kernel]

🫠

which set u got

[pure cedar]

3

.close

[devout snow]

Channel closed

Closed by @pure cedar

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[hallow echo]

hello, this is a question i had on a math competition for 2nd grade:
Let a,b and c be whole numbers and a≠0. What values can the discriminant have of the quadratic function:
f(x)=ax²+bx+c?

[tepid kestrel]

How is this 2nd grade

[hallow echo]

highschool

[tepid kestrel]

2nd grade in high-school? Educate me please

Anyways first intuition should be to try to make it a perfect square

[uncut crow]

i'm not sure i understand the question

[pure cedar]

no extra conditions? , then it can take any integer supposedly

[candid maple]

there's no restriction on the values of the polynomial, its roots, or its coefficients other than a != 0?

[tepid kestrel]

No I'm sure this one needs nt since they said abc are whole. Numbers probs some condition .

[solemn hatch]

Radioactive 2nd graders wtf

[uncut crow]

as stated it can take the form of anything that looks like b^2 - 4ac with a,b,c natural numbers and a not 0. how else would you describe it

[hallow echo]

well nothing else man

i saw the answer had like 7 different possibilities??

[pure cedar]

you can create any integer tbh
so i guess the answer must be set of all integers?

[pure cedar]

huh?

[tepid kestrel]

No you can't yk

[hallow echo]

hold on imma check the answer

[solemn hatch]

Idts

[uncut crow]

you can't make every integer

[solemn hatch]

It can't be 7

[uncut crow]

for example integers that are 3 mod 4 can't be made

[solemn hatch]

Yes

[pure cedar]

dang

nvm

[hallow echo]

the discriminant can have any value from 4Z U 4Z + 1 where 4Z = {4k : k ∈ Z} and 4Z + 1 = {4l + 1 : l ∈ Z}.

[uncut crow]

b^2 - 4ac is either 0 or 1 mod 4

[hallow echo]

thats the answer

[tepid kestrel]

Yeah these just use nt

[pure cedar]

oh ofc , quadratic residues

[tepid kestrel]

Use congruence to get restrictions then use euclids division lemma and prove or disprove the like 4 or 5 cases you get

[last parrot]

Bro have a nice pfp parth

I thought you western

[solemn hatch]

Yeah you can start with b=1,a=1,c=1

[pure cedar]

thanks , im always looking at the one whos typing

[pure cedar]

im indian

[last parrot]

Damn bro

[solemn hatch]

Then increase c by 1 to get all 1 mod 4

[hallow echo]

let me send the answer and imma translate it

so

it says:

[solemn hatch]

And b=0 for 0 mod 4

[hallow echo]

the D of the quadratic function is D=b²-4ac

[tepid kestrel]

Op are you from greece

[hallow echo]

who me?

[tepid kestrel]

Yes

Nvm

I'm tweaking

[pure cedar]

a simpler way to think is , the 4ac term will always be 0 mod 4
and b^2 can only be 0 or 1 mod 4

[solemn hatch]

No but how do we confirm it covers every one of them

I get that

[pure cedar]

covers everyone what?

[solemn hatch]

Every residue 0 or 1

[pure cedar]

i dont get what u are saying

[tepid kestrel]

Mod 4 has a property where if it's a perfect square it's 0 or 1 mod 4

[uncut crow]

every integer that is 0 or 1 mod 4 he is saying

[pure cedar]

oh

[solemn hatch]

We can confirm negative ones

[pure cedar]

there arent any negatives

only whole numbers

[solemn hatch]

You can make a negative D

[uncut crow]

discriminant can be negative

[tepid kestrel]

No y can take negative but d cannot

[pure cedar]

im talking about inputs

[tepid kestrel]

I mean can

[solemn hatch]

I was talking about D

[pure cedar]

i mean , we include negatives too when we represent em in mod
like -3 = 1 mod 4

[solemn hatch]

Right got it

Every residue 0 or 1

[tepid kestrel]

Op can you translate the question ty

[solemn hatch]

No but

Nvm

[pure cedar]

:3

[tepid kestrel]

No that's a congruency

Nit the discriminant

[solemn hatch]

Yes

[tepid kestrel]

Or im just dumb and not on the same boat yk

[pure cedar]

discrimination can be negative too , since no condition on roots is given in question

[solemn hatch]

But yes you can start b^2 from anywhere and start substraction by 4 for every residue

So it covers every residue

[tepid kestrel]

Wait

Yeah

I said cant

Wait I'm so dumb

[pure cedar]

thats not how mod works tbh
the whole purpose of mod is to eliminate invariants like 4ac

[solemn hatch]

Fam you are not getting me 😭

[pure cedar]

explain pls

[solemn hatch]

Mod won't give you the answer to if every natural number is covered

Which is in the form 4n+1 or 4n

[pure cedar]

it will duh

[solemn hatch]

You'll have to check that yourselves

Batao

[pure cedar]

when we say x = 1 mod 4

[solemn hatch]

Yaar choddo aap jee karo 🥀

[pure cedar]

we mean x can be anything of the form 4k + 1

so it covers every number of that form automatically

we don’t literally have to check stuff like a baby

[pure cedar]

thats not very polite tbh

[solemn hatch]

That's not either

[tepid kestrel]

whts JEE!
a sius
or Something

[solemn hatch]

If there is no other restriction on x then yes

[tepid stag]

entrance exam for colleges in india after 12th grade

[pure cedar]

so whats the issue then

[solemn hatch]

Here x is in the form b^2 - 4ac where b,c are whole numbers and a is a natural number

[pure cedar]

yea so

[solemn hatch]

That is condition imposed

[tepid kestrel]

Oh

[solemn hatch]

So you have to check

[hallow echo]

alr so

[tepid kestrel]

Is it easy or tough

[pure cedar]

nuh

[tepid kestrel]

Like sat for India

?

[hallow echo]

has anyone looked at the answer I've sent to explain it a bit further

[solemn hatch]

Alr gng

[tepid kestrel]

Nvm

[solemn hatch]

No

Sat is like cuet

JEE is career specific

[tepid kestrel]

Do you know congruence?

[pure cedar]

its irrelevant if a b c are naturals or whole
works the same in any case

[hallow echo]

lemme check

[uncut crow]

i'll write a self contained answer in one message.

b^2 -4ac is not 2 or 3 mod 4. the only possibilities are 0 or 1 mod 4. now we ask, for every integer M that is 0 or 1 mod 4, do there exist natural numbers a,b,c such that b^2 -4ac = M? yes, because b^2 can be arbitrarily large, then just consider the values b^2 -4, b^2 - 8, and so on.

[tepid kestrel]

Oh is it wasy

[hallow echo]

nnno

[solemn hatch]

Yupppp

That's what I've been saying

[tepid stag]

JEE is entrance exam for engineering colleges

[tepid kestrel]

So for the question to be done easily congruence are needed

Or you could use edl

[hallow echo]

okay can you explain it or i gotta find out myself

[solemn hatch]

b^2 is 0 or 1 mod 4, is it every natural number that is 0 or 1 mod 4 ?

[tepid stag]

there are others aswell like
NEET is for medical colleges
UPSC is for civil services or smth idk

[tepid kestrel]

You could read it in a few books

I'll send you one rq

[hallow echo]

alright

[pure cedar]

so ur saying the value of discriminant can be every natural number?

[solemn hatch]

Yes

[pure cedar]

nuh

ur wrong

[solemn hatch]

Yes

[uncut crow]

every integer that is 0 or 1 mod 4

[hallow echo]

btw here's the answer once more

who needs it

[solemn hatch]

When I try to ragebait but I'm mentally regressed

[pure cedar]

its okay

practice questions from pathfinder

[solemn hatch]

😭😭😭

[pure cedar]

youll get good in some time

[tepid kestrel]

You can read the nt part

[solemn hatch]

Sachin sir se padho pee dabloo mai 😈

[tepid kestrel]

Number theory

[pure cedar]

idk who that is

💔

[solemn hatch]

Good for you

[tepid kestrel]

Wait i think I know pathfinder

[solemn hatch]

Oh pathfinder is famous

[tepid kestrel]

I got one at a thrift shop

[solemn hatch]

Lmfao

[tepid stag]

!nopdfs

[devout snow]

Please post images (such as PNGs or JPGs) of the question rather than other filetypes such as PDFs which have to be downloaded. Non-image downloads can potentially contain viruses or other security risks.

[pure cedar]

its the best book for elementary olympiad prep

[tepid kestrel]

Oh cmon

[solemn hatch]

My teacher was one of the authors of that book

[tepid kestrel]

Nice

[pure cedar]

who

[hallow echo]

alright

[tepid kestrel]

I have the physics one

[solemn hatch]

Sachin singh

[pure cedar]

thats brainrot tbh

[pure cedar]

which pathfinder are you talking about

[solemn hatch]

Physics

[tepid kestrel]

I find it enjoyable to solve in my free time

[solemn hatch]

Maths pathfinder is ass wdym

[tepid kestrel]

It's moderate

[pure cedar]

its peak bro wth??

[pure cedar]

thats bshit tbh

[solemn hatch]

Nah way better are available wdym

[tepid kestrel]

Irodov is much betterimo

[solemn hatch]

It's enough for advanced, falls short in olympiads

[pure cedar]

which ones for example

[tepid kestrel]

My local community college uses it

[solemn hatch]

Bartle sherbert

[pure cedar]

i dont like physics that much tbh

[solemn hatch]

CPTCM

Problem solving strategies

Even tomato

[tepid kestrel]

It's not a good book yk

If the one ur saying is by Arthur engel

[hallow echo]

thank you mr document uploaded successfully

[solemn hatch]

It was tbh

Good for beginners

[tepid kestrel]

No...

[pure cedar]

i did a bit of mathematical circles
excursion in math
Sl loney (geometry)
path finder
evan chen
david m burton

[solemn hatch]

Why exactly?

[pure cedar]

im gonna do that now

[solemn hatch]

Evsn chan is nice

[tepid kestrel]

It doesn't cover major topics like complex geometry, invariant, group theory

[solemn hatch]

Mathematical circles is good too

[pure cedar]

yeahh but i dont like euclidean geometry that much so ..

[solemn hatch]

It's gimmicky in nature

It's not a classroom textbook

[pure cedar]

ok i gtg study

i hope its solved

[tepid kestrel]

Evan chen otis programme is good

[pure cedar]

!done

[devout snow]

If you are done with this channel, please mark your problem as solved by typing .close

[hallow echo]

sooo are we done

[tepid kestrel]

Ye

.clos3

[hallow echo]

ok what do i need to know yo solve something like this

[tepid kestrel]

.close

[hallow echo]

to*

[tepid kestrel]

Number theory

Alhebra5

Algebra

[hallow echo]

okay can i send the next question here on in a nee channrl

channel*

alr ill make a new channel

.close

[devout snow]

Channel closed

Closed by @hallow echo

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[devout snow]

Available help channel!

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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[violet pagoda]

B

[violet pagoda]

You cant simply this right?

[green crypt]

what is g(x)?

[violet pagoda]

Since X is Different from g(x)

[violet pagoda]

Not given

no g(x) is given

[stone stump]

!original

[devout snow]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[violet pagoda]

Its the original

[green crypt]

i gotta learn these commands

yeah just simply apply limit in this case

[stone stump]

well there is something cut off at the start

[green crypt]

i think thats just e)

[violet pagoda]

You cant put x in g(x) since x is not g(x)?

[stone stump]

which would imply there are parts a-d

and maybe even text before that

[green crypt]

you can put the argument like f(2)

[green crypt]

that is true perhaps its given at the start

[violet pagoda]

But you cant simply the limit

[green crypt]

ren can you send the question screenshot?

[stone stump]

if g is continuous then you can write it as 5 g(5)^2

if not then you cant do shit

but you arent giving us the whole context

[green crypt]

just send the whole page

lets peer pressure him into sending us it

[violet pagoda]

The question is different im just asking if what it is without the value of g(x) igven

[stone stump]

but given that whoever wrote this also wrote x=5 under the limit, maybe the question is just bad

[green crypt]

spoken like a true mathematician

[stone stump]

well yes if you dont know shit about g then of course you cant do shit

[violet pagoda]

If no g(x) given or value of it, then its final answer is Lim 5[g(5)]^2

[stone stump]

no, you need to know that g is continuous for that

[violet pagoda]

You cant simply it right?

Since x is not G(x)

Theres no value of g

You cant make it 125

[stone stump]

no of course you cant

g(x) could be anything

[violet pagoda]

Like this

look at 1

But they said its -162

[stone stump]

well either is given somewhere that g(x)=x or they fucked up

[green crypt]

facebook.....

[last parrot]

Too easy just write 6(g(-3))^3

[stone stump]

assuming g is continuous

[last parrot]

If no gx given then nothing can do atp

[devout snow]

@violet pagoda Has your question been resolved?

[harsh stream]

You mean if g is continuous at the point x=5

[devout snow]

Channel closed

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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[viral lynx]

I know I need to use this formula but does it matter which vector I take for y and which one I take for u? Or will they both get me the same result?

[mystic scarab]

Here y is the vector you want to project, so it must be [1 7]

And u is the vector on which you are projecting

[viral lynx]

Ah ok, makes sense

Thank you!

❤️

.close

[devout snow]

Channel closed

Closed by @viral lynx

Use .reopen if this was a mistake.

[mystic scarab]

Because the hat over y means the "new" y, hence it refers to the projected version of the original y

[devout snow]

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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[odd grove]

Hello!, so my question is on transformations (in functions) and this was a test question (not my answer my friends) and I am having trouble understanding how he got this answer

[odd grove]

the pic needs a second to load one sec

specifically the vertical compression by 1/4

[winter torrent]

it's to clear out that -4

[odd grove]

ohh

[winter torrent]

although i think it's incorrect for the record

[odd grove]

can u per chance explain why 😭 and whendoes it even apply

[odd grove]

yeah it doesnt seem right

so im rlly confused

[winter torrent]

you'd get like f(x-3) + 5/4 or something

which seems suboptimal

what i would do is start with a vertical shift of -5 to get rid of that "+5"

[odd grove]

i think he did that (11 units down)

which looking at it makes better sense 💔

[winter torrent]

yeah but like

that's not going to do what he wants

[odd grove]

ohhh

okay

[winter torrent]

order matters here

when doing graph transformations, your options are (with any number):

• multiply the whole frickin thing by -12
• add +8 at the end
• replace x with (3x)
• replace x with (x + 7)

[odd grove]

so it would be

• reflect on the x-axis
  -reflect on the y axis
  -vertically compressed by 1/4
  -horizontally compressed by 1/2
  -horizontally translated 3 units to the left
  -horizontally trsnslated 2 units left
  -vertical translated 5 units down
  -vertical translation 6 units down

[odd grove]

where does the -12 come from? (and other numbers)

[winter torrent]

you can use whatever numbers you want

[odd grove]

ohh

okay

[winter torrent]

the point is that those are the only things you can do

so like if we start with "reflect on the x axis" that means "multiply the whole heckin thing by -1"

[odd grove]

ohh

[winter torrent]

and we get -1 * ( -4 f(x-3) + 5 )

which is 4f(x-3) - 5

[odd grove]

so what we are tryna do is disolve the first equation?

[winter torrent]

mmmm our target is that k(x) thing over there, f(-2x + 2) - 6

but it seems simplest to start by trying to get to f(x)

if we can

[odd grove]

ohh okay okay

i dont think my teacher taught us that or i missed sum 😭

[winter torrent]

well... there are a few ways to do it. do you remember what your teacher taught?

[odd grove]

but so if im answering it should look like this right?

[winter torrent]

in that it's a list of steps, yes

[odd grove]

if im being honest not rlly

i look at yt vids to re-learn

and pray

i can find my class notes tho

[winter torrent]

mmk, well it's kinda like a puzzle game

[odd grove]

i think i understand how to do this tho but idk when i shoud apply this

[winter torrent]

like what is this used for?

it really becomes helpful when working with trigonometric functions like sin

[odd grove]

no no like, what questions need me to do this for future tests

OH

okay wait i think i understand, is it to equations that do not have any applied transformations correct?

[winter torrent]

um well sometimes you'll start with like √x and describe how to transform it to get to 3√(4x - 7) + 12

[odd grove]

oh yeah ik how to do that but i meant the disolving part specifically

[winter torrent]

just the same in reverse tbh

[odd grove]

ohh okay :3

THANK U SM FOR HELPING ME 😭

🙏

i rllu apreceate that

ima close this but pls have a wonderful day :3

.close

[devout snow]

Channel closed

Closed by @odd grove

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[normal bison]

any approaches ? also what should we keep in mind to solve it , i mean what exactly are we looking for to get the right equations to find the range

[winter torrent]

this is a bizarre question. are you allowed to choose more than one answer?

[coarse flume]

Is this true false or multiple choice

[winter torrent]

it's completely symmetric in x, y, z so it seems like if (A) is true then (B) is also true

[restive river]

so either AB or CD

[glossy dew]

take z = 4 - x - y and sub in the second equation

[normal bison]

yess

[glossy dew]

rearrange to form a quadratic in x and evaluate D >= 0 to find range of y

[normal bison]

its multiple correct

[glossy dew]

should work

[normal bison]

how did u conclude that just by the fact that it is symmetrical

[winter torrent]

because (A) and (B) are the same except we replaced x with y

if something is true for x then it must be true for y as well

since nothing in the problem distinguishes between them

[onyx torrent]

It can't be CD though cause if one of the numbers were close to 3 the the square would be greater than 6

[normal bison]

ohh

[winter torrent]

we could call our variables 🐖 and 🧀 instead

[normal bison]

we will get approx 5 min to find the answer . is there any shorter way ?

[glossy dew]

short enough

like 2 mins

[normal bison]

but then we will just get the range of x right

and by symmetry

we will say that

y and z will also have same range

?

right ?

[glossy dew]

yes

[normal bison]

ohhkk

[devout snow]

@normal bison Has your question been resolved?

[devout snow]

Channel closed

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Available help channel!

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Remember:
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• Show your work, and if possible, explain where you are stuck.
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• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[rain coral]

here

[rain coral]

help

i tried sketching it to help

but for c

any idea why for c if u cross normal and l1 direction vector and then cross it with normal again u get direction vector of p2

l2*

cos if u cross normal and l1 u find the line thats 90 degrees to both so thats like idek

(purple)

[harsh stream]

What have u done so far

[devout snow]

@rain coral Has your question been resolved?

[rain coral]

a

and b

[harsh stream]

So u need help on c right what have u done so far

For part c

[rain coral]

erm

nothing

cos

u cant really do anything without the idea

i got the position vector for l2

jst gotta find direction vector

i jst know

ur gna have to cross product things

but

[harsh stream]

Ok well no u dont

[rain coral]

id rather have the understanding

[harsh stream]

U can do it with dot prod

[rain coral]

well yh

but

[harsh stream]

And all the information they have given

[rain coral]

why wld u dot product

that wld give u a scalar answer

[harsh stream]

Yea but u can use those scalers to get the vector eq

Ok so lets do this

First let direction vector of L2 be V =(a,b,c)

[rain coral]

yh

[harsh stream]

Consider the dot prod of this with the normal of plane right should be 0 correct?

[rain coral]

yh

[harsh stream]

What did u get

[rain coral]

and dot product with l1 shld be the same as the plane

so do u use simulatenous eq?

[harsh stream]

Uh no

[rain coral]

cos the magnitudes will be in both equations

[rain coral]

didnt acc do it

jst understood

[harsh stream]

Ok first lets just do the dot prod with the normal bro

Go ahead

Tell me what ur equation is

[rain coral]

n=2,4,-1 and l2d=a,b,c so ur gna get 2a+4b-c/root21 x root a^2+b^2+c^2 =1

so

[harsh stream]

(a,b,c) dot prod the normal of the plane

[rain coral]

yh

thats it

[harsh stream]

U should get 2a+4b-c=0

Right?

[rain coral]

how come

dot product is sin no?

sin 90=1

[harsh stream]

The direction vector of l2 dotted with the nornal of plane is 0

Remember l2 lies inside the plane

[rain coral]

yh

[harsh stream]

Hence the normal dot with the direction of l2 is 0

Ok lets make it clear tho

,tex 2a+4b-c=0

[rain coral]

my fault

[woven radish]

Larper(Scholar Of Mysteries)

[rain coral]

my fault

nv,

nvm

nvm

yh igy

[harsh stream]

Next

[rain coral]

yh yh

and then dot the l2 and l1 and make it equal 10.6

[harsh stream]

Do yk the formalae for angle between a line and a line?

[harsh stream]

No

[rain coral]

angle between both is 10.6

so like theta is 10.6

[rain coral]

cross product

[harsh stream]

Cos(theta) = direction of l1 dot direction of l2 / mag(l1)* mag(l2)

We know theta from part b correct?

[rain coral]

yh thats dot product no?

[rain coral]

yh

10.6

[harsh stream]

No this is the formulae for angle between two lines

The question says the angle is theta and we found theta from part b?

[rain coral]

yes

[harsh stream]

From this make c the subject

Tell me what u get

[rain coral]

c=2a+4b

[harsh stream]

Ok now sub in to this

And leave it dont do anything with it

[rain coral]

so cos(10.6)= 4a-2b+7(2a+4b)/root 69 x root(a^2+b^2+(2a+4b)^2)

[harsh stream]

a and b are what we call free variable meaning we can choose the values lets pick nice numbers tho a =7 and b=-1 then c = 10

[harsh stream]

Now check theose numbers in this

[harsh stream]

Were allowed to do this remember this is from the plane

[rain coral]

wdym

[harsh stream]

A plane spans everything right

this means every point in the plane from what we got has coordinate (a,b,2a+4b)

Choose any values for a and b

Then c is forced to be 2a+4b

[rain coral]

yh

yh i understand

so

[rain coral]

wait whats the point

of this

[harsh stream]

To verify the points we picked satisfigy this condition

Satisfy

[rain coral]

whats the condition

[harsh stream]

The angle between the lines

Is theta remember

[rain coral]

yh

[harsh stream]

Or you could do cos^2 = 1 - sin^2 and we know sin^2 remember from part b

U should solve for a i did the working and ud get a =-7b

So now we have two conditions
C=2a+4b
a=-7b
So pick any b then a and c are forced

Does this make sense?

So basical b is an real number once u pick b then a is forced and c is forced

And it satisfies both conditions bc we got them via each condition right

[rain coral]

idk

after we made the assumptions and stuff

idk

i jst

didnt get it

the cross product way was in the ms and was 2 lines only

[harsh stream]

We didnt make any assumptions

We used the fact cos^2 =1-sin^2

Solved this we get a=-7b

So now weve forced a condition on a and b

[rain coral]

yh

i see

[harsh stream]

Btw ur right cross prod is faster

With works whatever u like more

[rain coral]

yh

i jst didnt get

the cross product

considering its alot faster

i jst

idk

cross product

always finds

the normal

thats the thing in the ms

but like

how dyk

like

the outcome of ur cross product lies on the plane or not

ykwim

[frail shore]

Hello

[devout snow]

@rain coral Has your question been resolved?

[rain coral]

hello

[lunar forum]

hello

[devout snow]

@rain coral Has your question been resolved?

[devout snow]

Channel closed

Closed due to timeout

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[devout snow]

Available help channel!

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[vast wyvern]

Let $G$ be a group and $G/Z(G)$ be cyclic. Show that $G = Z(G)$.

[woven radish]

ILikeMathematics

ILikeMathematics

ILikeMathematics

[vast wyvern]

We want to show that gh = hg for any h in G

[woven radish]

ILikeMathematics

[vast wyvern]

Oh wait

h = p^mZ(G)

So hg = p^(m + r)Z(G)

Yeah ok

.close

[devout snow]

Channel closed

Closed by @vast wyvern

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[devout snow]

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Remember:
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• Show your work, and if possible, explain where you are stuck.
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[cerulean ruin]

c) Let $d$ be any set. Prove there is an ordinal $\alpha$ such that $\alpha \notin d$, yet every $\beta < \alpha$ is an element of $d$.

I am a little stuck on this

[woven radish]

toast

[cerulean ruin]

\textbf{c) Let $d$ be any set. Prove there is an ordinal $\alpha$ such that $\alpha \notin d$, yet every $\beta < \alpha$ is an element of $d$.}
\\
Let $\alpha$ be any ordinal. Let $d = \bigcup\alpha$. I will claim that $\alpha \notin d$ but for all $\beta < \alpha$, $\beta \in d$. Suppose $\alpha \in \bigcup \alpha$ by way of contradiction. That means, there exists some set $y \in \alpha$ such that $\alpha \in y$. But because ordinals are transitive, we get $\alpha \in \alpha$. But by a lemma, this is a contradiction. Now assume by way of contradiction, there exists some $\beta^* < \alpha$, but $\beta^* \notin d$. Observe that if $\beta^* < \alpha$, then $\beta^* \in \alpha$.

[woven radish]

toast

[cerulean ruin]

I don't think $d = \bigcup \alpha$ is a valid set in this case right?

[woven radish]

toast

[cerulean ruin]

or am i on the right track

I think this will be true for most ordinals, but not all ordinals

wait hmm

wait

am i dumb

why cant we just use $d = \alpha$

[woven radish]

toast

[cerulean ruin]

cuz $\alpha \notin \alpha$ but every element $\beta \in \alpha$ is in $\alpha$?

[woven radish]

toast

[obsidian apex]

how do you know how to write in LaTeX?

just wondering

[devout snow]

@cerulean ruin Has your question been resolved?

[cerulean ruin]

just do it alot

<@&286206848099549185>

is this trivial

😭

[violet wind]

d isn't an ordinal though

I'm confused

[cerulean ruin]

uh

[violet wind]

You keep talking about letting d be this or that

But you don't get to choose d

[cerulean ruin]

oh ok

so we arent really constructing d

but then does d have to contain ordinals 0 through beta?

or is it saying

hm

[violet wind]

Well it doesn't have to by default

You have to find an alpha such that it does

[cerulean ruin]

so we pick any random d, then find some alpha such that any elements less than alpha is in d but not alpha itself

[devout snow]

@cerulean ruin Has your question been resolved?

[devout snow]

@cerulean ruin Has your question been resolved?

[twilit pebble]

You know you can ping helpers if they don't respond in 15 mins

[cerulean ruin]

i did 🗿

<@&286206848099549185> hey how do i approach this problem 🙏

[frosty crescent]

more set theory

[cerulean ruin]

yes

i will be here for half a year

🤣

[frosty crescent]

lol

[cerulean ruin]

but yeah

idrk how to appracoh

cuz like d can be any set

[frosty crescent]

lol, transfinite induction?

if 0 isnt in d then a = 0

if it is, induction time

[cerulean ruin]

oh hm...

[worn salmon]

D is any ordered set, more specifically

[frosty crescent]

d is any set

this holds for all sets

[worn salmon]

It has an order on it

[cerulean ruin]

wdym

[worn salmon]

If d is unordered then beta < alpha is meaningless

[frosty crescent]

but < is the epsilon relation for ordinals

[cerulean ruin]

well beta < alpha just means beta in alpha

[worn salmon]

Am I stupid?

[frosty crescent]

so < has a rigorous meaning

[worn salmon]

Must be

[cerulean ruin]

like d doesnt necessarily have to have an order

[frosty crescent]

do you know what an ordinal is

[cerulean ruin]

alpha does by definition

cuz ordinals are sets

or idk

[worn salmon]

I forgot what < means setwise, allow a foolish man his prattlings

[frosty crescent]

lol its fine

[cerulean ruin]

so this is solved with transfinite induction?

umm

let me think

[frosty crescent]

yeah i think

intuitively a is the smallest ordinal not in d

i would induct on epsilon probably

epsilon induction ftw

[cerulean ruin]

when it says every beta < alpha, does it imply 0 through beta must be in d

[frosty crescent]

yea

by definition

i would do epsilon induct

[cerulean ruin]

but what if we have like all finite ordinals, then our alpha is omega?

hmm

i see

but uh

[frosty crescent]

yeah

[cerulean ruin]

what if we have a set like

{1,3,5,7}

[frosty crescent]

buh just eps induct

[frosty crescent]

0

[cerulean ruin]

oh ok

[frosty crescent]

alpha = 0

[cerulean ruin]

and if its

{0,1,3,5,7} then alpha = 2?

[frosty crescent]

yeah

intuitively

[cerulean ruin]

ohhh

[frosty crescent]

alpha is the smallest ordinal not in the set