help-0

[cosmic scarab]

What i find more fascinating in this graph is the way it’s constructed, its really well made and gave me the perfect image from which i can expand on further

[prime badge]

there's no constant time solution

[lone heart]

@cosmic scarab Has your question been resolved?

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[lapis goblet]

do df/dx, by making e^y constant

then differentiate cosx

then do it again for the second derivative

do the same with df/dy

[lapis goblet]

basically trying to prove the 11 * 2 digit integer, shortcut trick
is it just enough to show that rhs is divisble by 11?

or just be like 11(100m + n), let 100m + n be some integer x, for -1010<x<1010, 11L = 11x,
Also another question am I allowed to do induction for this
Even tho it has negative integers, it still has a minimum value it can equal, and would it just be like all P(n), as long as it falls within the range

[gentle fossil]

can u tell thefinal ans

[lapis goblet]

no

try it yourself

[gentle fossil]

i dont know any differentiation

[lapis goblet]

?

answer is 0 @gentle fossil

now leave this channel please/ post in an empty channel

[gloomy grove]

What is the question tho?

[lapis goblet]

to prove that

its basically the 11* (any 2 digit integer) = some "shortcut" method

[gloomy grove]

We have to prove that 11L = 110m + 11n?

[lapis goblet]

ye

is it enough to just show RHS is divisble by 11

or like this: or just be like 11(100m + n), let 100m + n be some integer x, for -1010<x<1010, 11L = 11x,

[gloomy grove]

It's given than L = 10m +n

So just multiple 11 on both sides

[lapis goblet]

bro im actually dumb asf

i can just subsitute that in

3am moment

ty

.close

[lone heart]

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[faint hemlock]

Hello

[faint hemlock]

What is the c

In cayley Hamilton formula

I want to proof that

[mortal trellis]

the coefficients of the polynomial

[faint hemlock]

I know it but

[faint hemlock]

How I can get c

[faint hemlock]

Is c in matrix index?

[mortal trellis]

the c_i depend on the matrix entries but writing down a closed formula is a bit annoying

[faint hemlock]

.close

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[valid marsh]

Hey, I kinda need help with finding pattern in quadratic residues modulo some prime p

[valid marsh]

Is there a way to identify some what of a sequence for some prime p such that the sequence for any n produces a quadratic residue

[golden canyon]

I mean: a_n = (n^2 mod p)

or do you want them to be all different?

[lone heart]

@valid marsh Has your question been resolved?

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[whole bough]

I'm a total noob when it comes to math, however for my uni assignments I need to learn about line equations.
What are some good resources to learn about line equations and solve practice problems?
I would like to learn things like:

• finding coordinates of a direction vector given a line equation in the form of ax + by + c = 0
• finding line equation from 2 points etc.

[green juniper]

Idk if this will be of any use, but try this if you want?

this is just explanations

[lone heart]

@whole bough Has your question been resolved?

[whole bough]

This is definitely useful thank you. I would however also like to see something that has problems and examples

[lone heart]

@whole bough Has your question been resolved?

[nimble fern]

khan academy

[lone heart]

@whole bough Has your question been resolved?

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[solid cove]

Hello i would need help on this i forgot when the variables turn into ² and when theyre not i would just need help with this for now because i forgot how it works also im not that good in english so im not gonna be able to know every single word of the mathematics on english

[lapis goblet]

whats the actual question

like to combine them/ simplify them?

[solid cove]

What do i do at • and what to do at plus and minus i remember at one of them they turn into ²

[lapis goblet]

ok

[solid cove]

Simplify

[lapis goblet]

$\a\cdot a = a^2$

fuck

[solid cove]

Lol

[exotic canopy]

$a\cdot a = a^2$

[ocean seal]

artemetra

[lapis goblet]

that

[solid cove]

Yes

[exotic canopy]

$a\cdot a \cdot a = a^3$

[lapis goblet]

thats basiclaly hw to do the first one

[ocean seal]

artemetra

[exotic canopy]

and so on

[solid cove]

Theres a plus also in it can i just ignore that

No i cant

[lapis goblet]

keep the plus

[solid cove]

Oh

I get it

[lapis goblet]

first combine using the muliplication

then we'll get to the plus

[exotic canopy]

first do what's before the plus and after the plus

[solid cove]

u⁵ + u³

[exotic canopy]

mhm

[solid cove]

Alr

[lapis goblet]

thats it

[solid cove]

And what about the second one

[lapis goblet]

maybe its better to rearrange it so you have all the x's on one side and the y on the otherside

or you can just count how many x's there are

[solid cove]

Yes just mark both different colors makes it easier

[lapis goblet]

yeah or that

just make sure to include their signs

[solid cove]

But how do i simplify it

[lapis goblet]

well

how many x's did you count

[solid cove]

3

[lapis goblet]

ok

now

how many POSITIVE y's did you count

[solid cove]

2 and one negative

[lapis goblet]

so whats 2 -1

[solid cove]

1

[lapis goblet]

so its gonna be 3 positive x'sand 1 positive y

so +3x and + 1y

[ocean seal]

chef

[lapis goblet]

just getting rid of the uneeded stuff

do you get how thats the answer

[ocean seal]

chef

chef

[solid cove]

I do so when using minus and plus it cant turn into x³ but actually in 3x

[lapis goblet]

yea

mutliplcation with the same variable

makes it into exponents

[solid cove]

Because x³ is x•x•x

[lapis goblet]

when its just addition/subtraction

it will be coefficients/factors

[solid cove]

Thats just the easy stuff i forgot having a math test tmr so i bet ill ask way more questions today gonna try continuing with the harder stuff down there

[lapis goblet]

ye

when you're done with a channel

just remember to type ".close"

so someone else can ask for help later

then just open a new open channel when you need more help

[solid cove]

Well am i done i dont think so i still have these

Or should i close and ask if having problems again

[lapis goblet]

try then yurelf first

[solid cove]

Alr

[lapis goblet]

and if u get stuck then ask

peace

[solid cove]

Can i leave it open for now and ask if having problems

[lapis goblet]

i gotta go, but if someone doesn't answer ur new questions in like 15mins, u can ping "@ Helpers", and somene will get to u eventually

[solid cove]

Alr so i can keep it opened for now

[vague lark]

How’s it going

[solid cove]

E what do i do on d

[solid cove]

fractures always leave me thinking

Although i think that this ones easy

[vague lark]

You have a fracture?

[solid cove]

D

[vague lark]

Oh man

[solid cove]

What

[vague lark]

Collet the like terms

The ones with x and x^2 are different

[solid cove]

Im thinking the x²'s are -1/14 together

[solid cove]

So what i said is right?

[vague lark]

Did you work those out?

[solid cove]

I dont understand

What do you mean by that

Im german an easier word for it?

Easier sentence

And another questions what happens if there is no counter or what the name is left

[vague lark]

Did you answer the question I wrote in the photo?

[solid cove]

I know the red one

[vague lark]

Sorry I don’t understand

[solid cove]

-1/14x²

[vague lark]

Is it?

[solid cove]

I think so

[vague lark]

Wrong

[solid cove]

What

[vague lark]

Make the denominator the same

[solid cove]

Thats what i thought

[vague lark]

1/4 - 1/6 - 1/12
Becomes
3/12 - 2/12 - 1/12

Becomes 0/12

Becomes 0

[solid cove]

What

[vague lark]

0x^2 = 0

[solid cove]

So if theres nothing left above it just becomes zero

[vague lark]

Yes

[solid cove]

Yea got it now thanks been thinking about that but forgot if it was right

[vague lark]

Oh ok lol

[solid cove]

What about the x² does it stay or also disapear

[vague lark]

Disappear

Becases it’s 0

Now what’s -1/3 + 1/2

[solid cove]

Huh

Hm

First making it the same

[vague lark]

Make bottom number same

Yes

[solid cove]

So 1,5/3 x

[vague lark]

?

[solid cove]

No?

Its just the bigger

1/2 x

[solid cove]

Oh do i have to make both bigger

[vague lark]

No it’s just making the bottom number same

Fraction is still same

[solid cove]

So i still have to make both numbers bigger

I thought only of the one to match the others ones 3

[vague lark]

No

If you multiply bottom number with a number you have to do same with too number

[solid cove]

My math would be 1/2x times 1,5 equals 1,5/3x and then plus the -1/3x is 0,5/3x

Isnt that the same

I just made it harder?

[vague lark]

Yeah

[solid cove]

But it works?

[vague lark]

Yeah

0.5/3 = 1/6

[solid cove]

Alr but now i know the simpler thing

[vague lark]

So it’s 1/6x

[vague lark]

Nice

[solid cove]

Thank you

[vague lark]

Good progress

So what’s the answer then

[solid cove]

0 what we dont even have to write down 1/6x

0+1/6x 0 is not needed @vague lark

[vague lark]

No

It’s just 1/6 x

0 never needed as far as I know

Alright good job I got to go now bye

[solid cove]

Yes thats my point

Just saying

Its not needed

[vague lark]

Yes

Yes

1/6 x

[solid cove]

Alr

[vague lark]

Bye

Farewell

[solid cove]

Bye

I love the fact that im having problems with every single one of them because we didnt speak of that and started a other theme but its still gonna be in our class test

Now having problems with e

What do i do on e

<@&286206848099549185>

Man

Someone

[charred jewel]

just multiply everything

[solid cove]

How does it just turn into 20u⁵v⁴•24uv²

I dont get it what to multiply

And what not

[charred jewel]

whats 4 * 5 * 6

[solid cove]

120

If ur saying times

Idk what

Means

[charred jewel]

multiply

[solid cove]

Yea 120

[charred jewel]

now whats u * uv * v^2

[solid cove]

u²v³

[charred jewel]

then just put them together

so its 120u²v³

[solid cove]

So whem multiplying it doesnt matter wich variable it has i can just multiply whatever i want

I dont understand any of them

As nd what do i do on the next one

And

On f

[charred jewel]

f, distribute the power then multiply all of them

[solid cove]

What does that mean im german

Distribute

[charred jewel]

means give every number and variable in the brackets the power

[solid cove]

It has ()⁴

[charred jewel]

yes

[solid cove]

I dont get what i do there

I know how to multiply it now

But not what to do when that happens

[charred jewel]

ok so you have (5r³s²)⁴

[solid cove]

Yes

[charred jewel]

raise every number and variable in the brackets to the power of 4

[solid cove]

R⁷s⁶

20

No?

[charred jewel]

no

you multiply the powers not add

[solid cove]

500

[charred jewel]

5^4 = 500?

[solid cove]

Yes

[charred jewel]

you sure?

[solid cove]

5times 5 20 times 5 100 times 5 500

Isnt that how to

Do the 5⁴

[charred jewel]

5 times 5 20?

[solid cove]

Oh

Im really smart

Am I

I am very smart really no lies

💀

So its not 500 mb

[charred jewel]

nope

[solid cove]

625

[charred jewel]

ok good

[solid cove]

But what about the variables

Its different there

[charred jewel]

you also give them the power of 4

whats (r³)⁴

[solid cove]

I dont know how to solve that

Thats the problem

There

I dont know what to do with them

[charred jewel]

well you know what is 2³?

[solid cove]

8

No

[charred jewel]

you multiply by 2 3 times yes?

[solid cove]

What

16

[charred jewel]

yes it is 8

[solid cove]

A

[charred jewel]

2³ means you multiply 2 three times right?

[solid cove]

2times2times2

[charred jewel]

yes

[solid cove]

Mb

[charred jewel]

now apply same logic

[solid cove]

But any variables are 1

[charred jewel]

what is (r³)⁴

no?

[solid cove]

Whatttt

[charred jewel]

variables dont have a value unless you give them one

ok so (r³)⁴ means you multiply r³ 4 times yes?

[solid cove]

is it basically 3times3times3times3

[charred jewel]

yes

[solid cove]

Bruh

81

Is it really r⁸¹

That cant be

[charred jewel]

no..?

[solid cove]

What

What am i doing wrong

[charred jewel]

wait no shit

wrong

whats 4 * 3

[solid cove]

But

Dont i just add them

On variables

Would (r³)⁴ just be r⁷

[charred jewel]

no

[solid cove]

Thats how i remember

[charred jewel]

thats if you have r³ multiply r⁴

[solid cove]

So lets say it is r¹²

Why do i multiply it

Not add it

[charred jewel]

ok so do you agree that (r³)⁴ = r³ * r³ * r³ * r³

[solid cove]

No

[charred jewel]

ok do you agree that 2³ = 2 * 2 * 2

[solid cove]

Wait i get it now

im not multiplying it its adding them all together

Thanks dude

[charred jewel]

heres a table to help you on your problems

[solid cove]

Never had most of that stuff

Or

Sunno

Dunoo

[solid cove]

But i got it now thank you so much

[lone heart]

@solid cove Has your question been resolved?

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[thick lynx]

(Without having its formula in the first place, I mean, how would you derive it)

[vale wigeon]

$\sum_{k=1}^n (k^3 - (k-1)^3) = n^3$

[ocean seal]

AnnGhost

[vale wigeon]

k^3 - (k-1)^3 is a quadratic in k

tinker with that

[stable sierra]

i think we have to prove it equal to 2n+1(n+!)(n)/6

[thick lynx]

I'm asking how we would derive that expression in the first place, not just proving it is equal to that using complete induction

[thick lynx]

Oh

[thick lynx]

Hm, we would have to make it just k^2.
We can break it up, since we know Gauss' formula

[keen plinth]

,w expand 1/3((k+1/2)^3 - (k-1/2)^3) - 1/12

[ocean seal]

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[thick lynx]

,w expand k^3 - (k - 1)^3

3k^2 - 3k + 1

[\sum_{k=1}^n k^2 = \sum_{k=1}^n (k^3 - (k - 1)^3) = \sum_{k = 1}^n (3k^2 - 3k + 1)] [= n - \frac 3 2 (n(n+1)) + 3\sum_{k =1}^n (k^2).]

[keen plinth]

[keen plinth]

is this not more direct

[thick lynx]

Hm, is this what Ann meant?

[keen plinth]

[
\sum_{k = 1}^n k^2 = \sum_{k = 1}^n \parens {\f13 \parens {\parens {k + \f12}^3 - \parens {k - \f12}^3} - \f1{12}}
]

[thick lynx]

How did you come up with that

[keen plinth]

this

[thick lynx]

I mean the long term

That evaluates to k^2

[lean forge]

Geometric proof is pretty classic

[keen plinth]

yeah check the wolfram output

i just put in (k + 1/2)^3 - (k - 1/2)^3 and then fixed the error

[thick lynx]

oh

[keen plinth]

,w expand (k + 1/2)^3 - (k - 1/2)^3

[ocean seal]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by tearfox

[thick lynx]

Why + 1/2 and - 1/2?

[keen plinth]

because then its cleaner

as you can see

[thick lynx]

ok

[thick lynx]

@vale wigeon Is this what you meant?

[vale wigeon]

yes

[thick lynx]

Alright, thanks. Yeah, we can then solve for the sum of k^2

Thank you!

Or do it similarly with Snow's way

Thanks!

.close

[lone heart]

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[vagrant lance]

hey everyone! I hope u are doing well
I need help with that specific question. Idk the signification of the n+1 or the n written down side a and b. Can you please help meee .the exercise is in french

[vagrant lance]

PLEASE HELP MEE

[upbeat hornet]

It means it is the nth term in a sequence

[lean forge]

an+1+sqrt(2)bn+1 = (1+sqrt2)^(n+1) = (1+sqrt2)(1+sqrt2)^n

Replace (1+sqrt2)^n by its expression

[vagrant lance]

THank you sooo veryyy much Azenx

can u do the following question?

[lone heart]

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.

[upbeat hornet]

which one?

[vagrant lance]

c/ please

[upbeat hornet]

Can you translate the text of question 5 into English?

[vagrant lance]

it's verify that ...

the last one

[upbeat hornet]

The whole text

Because that’s where it defines a_n and b_n

[vagrant lance]

aah okay

It's clear I guess

[lean forge]

a / Show that : ...
b : write ... in function of ...

[vagrant lance]

5/
a/ demostrate that..

[upbeat hornet]

Consider the powers of (1-sqrt(2))^n

[vagrant lance]

b/ write in fucntion of

what about using a remarkabal identity

[upbeat hornet]

What identity?

[lean forge]

a²-b² = (a+b)(a-b)

[upbeat hornet]

Yes

This identity is the key

[vagrant lance]

okaaay

[upbeat hornet]

Use this identity together with this

[vagrant lance]

okay thank you so much guys !

ily7

[lone heart]

@vagrant lance Has your question been resolved?

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[reef basin]

yo

[reef basin]

can someone help me in that question

???

[limpid turret]

What is definition of reflexive

[reef basin]

Relation R is reflexive if xRx for every x ∈ A.
That is, R is reflexive if ∀ x ∈ A, xRx

and R is a subset of A*A

[limpid turret]

In this case, $R=\emptyset$

[ocean seal]

SWR

[limpid turret]

And it's helpful to write $xRx$ as the classical $(x, x)\in R$

[ocean seal]

SWR

[reef basin]

but what about the properties of R

[limpid turret]

what properties?

[reef basin]

this question demands whether the given relation reflexive, symmetric or transitive

[limpid turret]

Yes. And we're tackling reflexive first.

[limpid turret]

I was saying that, for reflexive, it's better to rewrite the condition as shown here

[reef basin]

ehe got it

but it is not reflexive right ?

as long as the condition doesnt met for all elements in A

[lone heart]

@reef basin Has your question been resolved?

[lone heart]

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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!

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[solid cove]

This shit doesnt make sense

[unkempt robin]

What doesn't make sense?

[solid cove]

I dont get what to do i dont understand anything

[unkempt robin]

Do you know about the distributive law or the binomial formulas?

[solid cove]

I do

These are all solutions from out teacher but i dont get how he got that

[unkempt robin]

The expressions are in the factored form. You are asked to expand them using either

[solid cove]

Our

[unkempt robin]

Use $a \cdot (b + c) = ab + ac$

[ocean seal]

USS-Enterprise

[unkempt robin]

(the distributive property) to solve the first one

[solid cove]

Bro where do i get a from

I dont get it

A just appears

[unkempt robin]

Imagine. You've got $(5a + 6b) \cdot (7c + 8d)$

[ocean seal]

USS-Enterprise

[unkempt robin]

the a, b and c I used are general

Think of (5a + 6b) as a

This a has nothing to do with the a in your example!

And think of 7c as b and 8d as c

And you end up in the form of $a \cdot (b + c)$

[ocean seal]

USS-Enterprise

[solid cove]

[unkempt robin]

a, b, c are some real numbers

In your case, a = 5a + 6b

But the two a's aren't the same

We can pick any letters

Say x, y, z

x = 5a + 6b, y = 7c, z = 8d

You are again in the form of $x \cdot (y + z)$

[ocean seal]

USS-Enterprise

[solid cove]

And what do i do with a

I cant just take two numbers

Do i have to solve a first

Then multiply

[unkempt robin]

You can't solve anything. It's an expression that you have to expand

[solid cove]

What

[unkempt robin]

$(5a + 6b) \cdot (7c + 8d)$

[ocean seal]

USS-Enterprise

[unkempt robin]

Is a mathematical expression

You can not solve anything here

[solid cove]

How does the teacher come to this solution then

[unkempt robin]

What you have to do however, is expand.

Which in this case means multiplying everything

[solid cove]

Oh

[unkempt robin]

Using the distributive property or binomial formulas

So

[solid cove]

I get the first one

[unkempt robin]

Okay

$(24u - 15v)^2$

[ocean seal]

USS-Enterprise

[unkempt robin]

What's this?

[solid cove]

Formula 1

[unkempt robin]

What

[solid cove]

Binomic

Idk how to translate it

Im german

[unkempt robin]

Ah

[solid cove]

Binomische formel 1

[unkempt robin]

Well we don't number them

This is a square of difference

You are squaring the difference of two numbers

Think of $(a - b)^2$

[ocean seal]

USS-Enterprise

[unkempt robin]

Where a = 24u, b = 15v

[solid cove]

Huh

[unkempt robin]

How do we expand this?

[unkempt robin]

In math you often have to imagine things

We have $(24u - 15v)^2$

[solid cove]

a-ab+b

[ocean seal]

USS-Enterprise

[unkempt robin]

If an unexperienced person looks at this they might get baffled

But if we say that a = 24u and b = 15v

We get $(a - b)^2$

[ocean seal]

USS-Enterprise

[unkempt robin]

And that's the square of difference

[unkempt robin]

No

[solid cove]

Whaa

T

[unkempt robin]

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

[ocean seal]

USS-Enterprise

[solid cove]

a²-ab+b²

[unkempt robin]

2ab

[solid cove]

Yes

Yup

[unkempt robin]

So

Then what is $(24u -15v)^2$

[ocean seal]

USS-Enterprise

[solid cove]

24u² -11uv+15u²

I dont know

[unkempt robin]

It's the first squared minus 2 times first times second plus second squared

The first is 24u, the second is 15v

[solid cove]

Man

My brain just cant

[unkempt robin]

Okay, let's try again

Let's say some random variable a = 24u

And some random variable b = 15v

We get $(a - b)^2$

[ocean seal]

USS-Enterprise

[unkempt robin]

Do you understand this?

[solid cove]

Yes

[unkempt robin]

Okay

And let's expand this

[solid cove]

a²-2ab+b²

[unkempt robin]

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

[ocean seal]

USS-Enterprise

[unkempt robin]

Correct

So we've got $a^2 - 2ab + b^2$

[ocean seal]

USS-Enterprise

[unkempt robin]

And we just get a and b back to what they are

[solid cove]

And how do i make the one with the numbers

[unkempt robin]

a = 24u, b =15v

[solid cove]

Thats what im struggling with

[unkempt robin]

Just replace all a's with 24u and all b's with 15v

[solid cove]

Nit that simple

U still have to do more

[unkempt robin]

Okay let's do it step by step

We have $a^2$

[ocean seal]

USS-Enterprise

[unkempt robin]

And if a = 24u, what is a^2

[solid cove]

24u²

[unkempt robin]

No.

[solid cove]

What

[unkempt robin]

We get $(24u)^2$

[ocean seal]

USS-Enterprise

[solid cove]

Thats what im saying

[unkempt robin]

or $(24u) \cdot (24u)$

[solid cove]

Almost

[ocean seal]

USS-Enterprise

[unkempt robin]

$24 \cdot u \cdot 24 \cdot u$

[ocean seal]

USS-Enterprise

[unkempt robin]

Using commutativity, $24 \cdot 24 \cdot u \cdot u$

[ocean seal]

USS-Enterprise

[solid cove]

(24u)²-11uv+(15v)²

[unkempt robin]

You have to square the number and the variable!

[unkempt robin]

Close

The middle part

Let's focus on this

$-2ab$

[ocean seal]

USS-Enterprise

[solid cove]

Oh

[unkempt robin]

Or, $-2 \cdot a \cdot b$

[ocean seal]

USS-Enterprise

[unkempt robin]

What do we get if we put back a and b?

[solid cove]

Wait let me think

[unkempt robin]

a = 24u, b = 15v

[solid cove]

37uv?

[unkempt robin]

Don't give me the result

Type it out for me

[solid cove]

wdym

[unkempt robin]

How'd you get 37?

[solid cove]

The whole

15-2= 13 +24= 37

[unkempt robin]

hm

But there's no addition or substraction here

You are multiplying -2 by a and then that by b

[solid cove]

wait i should multiply?

[unkempt robin]

-2 times a times b

if we get a and b back

-2 times 24u times 15v

Or $-2 \cdot 24u \cdot 15v$

[ocean seal]

USS-Enterprise

[unkempt robin]

the dot represenents multiplication

[solid cove]

-70uv

Wait

No

78

-78

Uv

[unkempt robin]

How'd you get that?

[solid cove]

Ehm

Wait i did -2(24u+15v

Mb

[unkempt robin]

That is incorrect

You've only got multiplication

[solid cove]

If i get -48u times 15v its -720uv

[unkempt robin]

There you go

So we've got the middle part

[solid cove]

Finally

[unkempt robin]

$(24u)^2 -720v + (15v)^2$

[ocean seal]

USS-Enterprise

[solid cove]

Yes

[unkempt robin]

Now just square the left and the right

[solid cove]

One sec gonna write down the other c first

And now to d

So

[unkempt robin]

Woah woah

We aren't done yet

What do you get after squaring?

[solid cove]

Wrote down the other one we just did not this one

576u²-720uv+225v²

[unkempt robin]

👍

Next one

$(3r^4+8)^2$

[ocean seal]

USS-Enterprise

[solid cove]

Looks easy

[unkempt robin]

Yep

What do we have

[solid cove]

wait ima do it rq

And youll see

I think i know how to now

9r⁸+48r+64?

[unkempt robin]

Close

It's not 48r though

[solid cove]

48r⁴

[unkempt robin]

There you go

[solid cove]

W

[unkempt robin]

$9r^8 + 48r^4 + 64$

[ocean seal]

USS-Enterprise

[unkempt robin]

Okay, f

[solid cove]

Let me write it down rq

[unkempt robin]

$(18p + 19q) \cdot (18p - 19q)$

[ocean seal]

USS-Enterprise

[unkempt robin]

Sure thing

[solid cove]

Cant i do something on e

Make it smaller

No i cant

Mb

[unkempt robin]

You can not

[solid cove]

Because r⁸ and r⁴ is something different mv

Mb

So

F

[unkempt robin]

$(18p + 19q) \cdot (18p - 19q)$

[ocean seal]

USS-Enterprise

[unkempt robin]

Do you notice something?

[solid cove]

324p²-342pq+342pq-361q²

[unkempt robin]

Yes

But what's -342pq + 342pq

[solid cove]

0

[unkempt robin]

There you go

Also, there's a quicker way to do this

Again, let's say a = 18p and b = 19q

We get $(a + b) \cdot (a - b)$

[ocean seal]

USS-Enterprise

[unkempt robin]

This is a difference of squares

$a^2 - b^2 = (a+b) \cdot (a-b)$

[ocean seal]

USS-Enterprise

[unkempt robin]

And a^2 = 324p^2, b^2 = 361q^2

So we get 324p^2 - 361q^2

[solid cove]

I dont get it ill just stick to the harder way

Now for the ultimate test g

[unkempt robin]

Should learn it

Will be very useful

Especially in the other direction

When you have $324p^2 - 361q^2$

[ocean seal]

USS-Enterprise

[unkempt robin]

And gotta get to the factored form

[sharp orbit]

how do you use TeXit that neatly?

[unkempt robin]

Helps to know that this is the same as $(18p)^2 - (19q)^2 = (18p + 19q) \cdot (18p -19q)$

[ocean seal]

USS-Enterprise

[unkempt robin]

Years of experience 🥱

I'm just joking, it's not that hard. Just practice 🙂

[sharp orbit]

nice : D

[unkempt robin]

Anyway, f

[solid cove]

Charles been stalking us

[unkempt robin]

GIve me a minute to write it in latex

[sharp orbit]

format this pls $(18p)^2$

[solid cove]

What

[unkempt robin]

$(114s - 78t) \cdot \left(\frac{s}{2} - \frac{t}{3}\right) - (19s + 63t) \cdot (3s -14t)$

[ocean seal]

USS-Enterprise

[solid cove]

Let me write down f

[unkempt robin]

First thing I'd do is look around a bit and realize there's nothing easy to do right away other than go straight into multiplying everything and getting rid of brackets

[solid cove]

This made my head explode again thank you

[unkempt robin]

I do notice however that we are substraction the multiplication of two numbers from another multiplication of two numbers

So, because of the order of operations

THis is the same as

[solid cove]

For me the g doesnt looks that hard

[unkempt robin]

OOps

[solid cove]

S/2 is basically 1/2s and t/3 1/3t

[unkempt robin]

$\left((114s - 78t) \cdot \left(\frac{s}{2} - \frac{t}{3}\right)\right) - \left((19s + 63t) \cdot (3s -14t)\right)$

[ocean seal]

USS-Enterprise

[unkempt robin]

There we go

Had the brackets in the wrong place

So

We have to first simplify the left parantheses and the right

Then substract

[unkempt robin]

Correct

[solid cove]

Cant we just give the 114 a 1/114

Same with 78t

[unkempt robin]

What?

[solid cove]

Cant i just write down 1/114

To make it easier

[unkempt robin]

uh

no?

[solid cove]

No

Alr

[unkempt robin]

$114 \neq \frac{1}{114}$

[solid cove]

I thought there was something like that

[ocean seal]

USS-Enterprise

[solid cove]

Who created this genius bot

[unkempt robin]

Geniuses

[solid cove]

True

[unkempt robin]

Anyway

[solid cove]

Hm

[unkempt robin]

$\left((114s - 78t) \cdot \left(\frac{s}{2} - \frac{t}{3}\right)\right) - \left((19s + 63t) \cdot (3s -14t)\right)$

[ocean seal]

USS-Enterprise

[unkempt robin]

There's nothing you can do instead of jumping right in and multiplying stuff

[solid cove]

How can i multiply the fractures

[unkempt robin]

Let's first split it into 2 parts

The left and the right parantheses

$(114s - 78t) \cdot \left(\frac{s}{2} - \frac{t}{3}\right)$

[ocean seal]

USS-Enterprise

[unkempt robin]

This is the left

Think of it as $(a - b) \cdot (c - d)$

[ocean seal]

USS-Enterprise

[unkempt robin]

Use FOIL

[solid cove]

I dont get how to multiply them

[unkempt robin]

What do you mean

It's the same thing we did in the start

[solid cove]

But its woth fractures

[unkempt robin]

And?

[solid cove]

Jow do i multiply them

How

[unkempt robin]

Okay

So we get $114s \cdot \frac{s}{2}$

[ocean seal]

USS-Enterprise

[unkempt robin]

You get this?

For the first

[solid cove]

Yes

I know what to do

But not how to

[unkempt robin]

You multiply the integer with the numerator

[solid cove]

Gonna have to translate rq

[unkempt robin]

Well

Let's say we had $a \cdot \frac{b}{c}$

[solid cove]

The thing above

[ocean seal]

USS-Enterprise

[solid cove]

We multiply b

[unkempt robin]

We solve this by multiplying a with the numerator (b)

[solid cove]

Alr

[unkempt robin]

$\frac{ab}{c}$

[ocean seal]

USS-Enterprise

[solid cove]

I get it

Got it

So

[unkempt robin]

So we've got $\frac{114s^2}{2}$

[ocean seal]

USS-Enterprise

[unkempt robin]

Gotta simplify this fraction

[solid cove]

Just take the half

[unkempt robin]

Yes

[solid cove]

57

1

[unkempt robin]

What

[solid cove]

No?

What

[unkempt robin]

What's the 1?

Oh you wrote it as a fraction

[solid cove]

We can turn this into 57/1

[unkempt robin]

Yes $\frac{57s^2}{1}$

[ocean seal]

USS-Enterprise

[unkempt robin]

But that's just 57s^2

You divided the top and bottom with 2

[solid cove]

Yes

[unkempt robin]

Same reason $\frac{8}{2} = \frac{4}{1}$

[ocean seal]

USS-Enterprise

[unkempt robin]

Okay, so do this for everything

$(114s - 78t) \cdot \left(\frac{s}{2} - \frac{t}{3}\right)$

[solid cove]

Alr 5mins

[ocean seal]

USS-Enterprise

[unkempt robin]

And then the right part

And then substract everything

[solid cove]

Got it

[unkempt robin]

👍

[solid cove]

57s²/1-38st/1-39st/1+26t²+38s²-765st+189st-882t²

I thank u alot for ur help but im dead now

Ima take a break for today

[unkempt robin]

Hang on

You're correct until +38s^2

[solid cove]

Huh

[unkempt robin]

You should have gotten $(57s^2 - 77st - 882t^2)$

[ocean seal]

USS-Enterprise

[solid cove]

Oh 77st

[unkempt robin]

In the second parantheses

[solid cove]

Mb

[unkempt robin]

Then substracted that

[solid cove]

What did i do wrong

[unkempt robin]

So $57s^2 - 38st - 39st +26t^2 -(57s^2 - 77st -882t^2)$

[ocean seal]

USS-Enterprise

[unkempt robin]

Not sure, try multiplying the thing inside the right parantheses again

$(19s + 63t) \cdot (3s -14t)$

[ocean seal]

USS-Enterprise

[unkempt robin]

This

Should end up with $(57s^2 - 77st -882t^2)$

[ocean seal]

USS-Enterprise

[solid cove]

I did 19times 14

Why doesnt that work

[unkempt robin]

Hang on let me try

$(19s + 63t) \cdot (3s -14t) = (57s^2 - 266st + 189st - 882t^2) = (57s^2 - 77st - 882t^2)$

[solid cove]

Huh

[ocean seal]

USS-Enterprise

[unkempt robin]

There

You use FOIL

[solid cove]

What did i do wron

G

[unkempt robin]

You multiply 19s with 3s, then 19s with -14t, then 63t with 3s, then 63t with -14t

[solid cove]

I dont get it

[unkempt robin]

$(a+b) \cdot (c-d) = ac - ad + bc - bd$

[ocean seal]

USS-Enterprise

[solid cove]

Still dont see my mistake huh

This just making me mad somehow

[unkempt robin]

¯_(ツ)_/¯

[solid cove]

Where the mistake

What

[unkempt robin]

I don't see your work so I don't know

[solid cove]

Im not gonna wonder if u cant read it

And the 57s² is there to i just accidentally cropped it out

[unkempt robin]

Not sure what you did there

[solid cove]

I also dont understand how to get that solution

[unkempt robin]

$(19s + 63t) \cdot (3s -14t)$

[solid cove]

How do i get the solution im supposed to what

[ocean seal]

USS-Enterprise

[solid cove]

I did that

Theres no thing where i can get a 7u

77

[unkempt robin]

$= (19s \cdot 3s - 19s \cdot 14t + 63t \cdot 3s - 63t \cdot 14t)$

[ocean seal]

USS-Enterprise

[unkempt robin]

$= (57s^2 - 266st + 189st - 882t^2)$

[ocean seal]

USS-Enterprise

[unkempt robin]

$= (57s^2 - 77st - 882t^2)$

[ocean seal]

USS-Enterprise

[solid cove]

How do u get that

[unkempt robin]

I literally posted the steps

From start to finish

[solid cove]

I dont get the steps

What

Why did it turn into smaller numbers

Suddenly

[unkempt robin]

You collect like terms

-266st + 189st

Same as -266x + 189x

[solid cove]

OH

[unkempt robin]

-77x or -77st

[solid cove]

oh

I get the mistake but too lazy too correct it thank you alot again

Gonna close it now and chill a lil

Bye

[unkempt robin]

Wait a bit

2 mins

[solid cove]

Why

[unkempt robin]

So we get $57s^2 - 38st - 39st +26t^2 -(57s^2 - 77st -882t^2)$

[ocean seal]

USS-Enterprise

[solid cove]

Yes

[unkempt robin]

$57s^2 - 38st - 39st +26t^2 - 57s^2 + 77st + 882t^2$

[ocean seal]

USS-Enterprise

[unkempt robin]

Again collect like terms

57s^2 - 57s^2 = 0

[solid cove]

Oh

It can get even smaller

I get it

[unkempt robin]

-38st - 39st + 77st = 0

[solid cove]

Yup yup thanks

[unkempt robin]

So we end up with $26t^2 + 882t^2$

[ocean seal]

USS-Enterprise

[unkempt robin]

Or $908t^2$

[ocean seal]

USS-Enterprise

[unkempt robin]

And that's the final answer

$\left((114s - 78t) \cdot \left(\frac{s}{2} - \frac{t}{3}\right)\right) - \left((19s + 63t) \cdot (3s -14t)\right) = 908t^2$

[ocean seal]

USS-Enterprise

[solid cove]

Bye

[unkempt robin]

Anyway, that's it

Have a good day

[solid cove]

Ty again you too

[unkempt robin]

No problem

[solid cove]

You helped alot

[unkempt robin]

🙂

[solid cove]

.close

[lone heart]

Channel closed

Closed by @solid cove

Use .reopen if this was a mistake.

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[kindred folio]

Let 𝑓: [−2, 2] → ℝ be a function defined by
expression 𝑓 𝑥 = 4 − 𝑥2 . The sum of squares
of the domain values ​​that have an image equal to
√2 is:
a) 2
b) 3
c) 4
d) 5
e) 6

[kindred folio]

ive studied linear functions before but im still a bit confused on the concept of domain and image, can somebody help me with this one?

[pastel bane]

Wait what is the function there? Is it f(x)=4-2x?

Or is it $f(x)=4-x^2$?

[ocean seal]

goobybalooby

[kindred folio]

oh it got messed up when i pasted wait

this is the function

[unkempt robin]

Well that's news

[pastel bane]

Oh okay

[keen mason]

well thats totall different

[pastel bane]

What is your intuition for the problem?

[kindred folio]

initially, because its a quadratic function i thought of finding the roots but it doesnt make sense

i just dont know what the "domain values" would be, and how do i know their image

[unkempt robin]

This is not a quadratic function.

[pastel bane]

Well all possible domain values are the real numbers in the closed interval [-2,2]

And yes, that is not quadratic with the square root

[kindred folio]

oh i thought it was bc of the x^2

[kindred folio]

the domain values are all the possible values for y?

[pastel bane]

It would be if there were no square root

The domain values are all of the possible inputs for x that you are “allowed” to use for a given function

In your previous statement $f: [-2,2] \rightarrow \mathbb{R}$

[ocean seal]

goobybalooby

[pastel bane]

The domain is the interval left of the arrow, the codomain is the real numbers right of the arrow, and the image is the “range” of the function, which is all possible values of f(x) when you vary the input x throughout the domain

[kindred folio]

so it means all the real numbers in the [-2, 2] interval?

[pastel bane]

Are you asking about the domain or the image?

[kindred folio]

that would be all the values x can be in the function

[kindred folio]

the domain

[pastel bane]

Yes it sounds like you’re understanding

This is just nailing down terminology tho, the problem itself involves just a tiny lil calculation if you fully understand the terminology

[kindred folio]

yes, i was just a little confused on the terminology

[pastel bane]

For sure. So how would you move forward from here?

[kindred folio]

okay, so if it wants the domain values that have an image equal to sqrt(2), do i have to calculate all of the "x values" in that interval that have their y value equal to sqrt(2)?

like sqrt(2) = * function * then find the 2 x values by finding the roots?

[pastel bane]

Yes

But their not called roots

[kindred folio]

yes, because they're not on the y = 0 coordinate

[kindred folio]

ok wait

[pastel bane]

You gotta solve $\sqrt{2}=\sqrt{4-x^2}$ for x

[ocean seal]

goobybalooby

[kindred folio]

x^2 = 2, so x = positive and negative sqrt(2)?

[pastel bane]

Yes

But recall the statement of the question: it’s looking for the sum of the squares of the domain values

[kindred folio]

so the sum of the squares would be 2 + 2? because the square of -sqrt(2) would also be positive 2

[pastel bane]

There ya go

Also, just to continue to nail down terminology, note that sqrt(2) is approximately 1.41 so that is in the domain [-2,2]

[kindred folio]

what if it wasnt in the domain? the result would be ∅?

because there would be no x values in the domain that have the y value equal to sqrt(2)

[pastel bane]

Yeah they’d have to change the list of possible answers to include 0 lol

[kindred folio]

yeah, one last question, what is the function if its not a quadratic function?

[pastel bane]

Well in this specific case, the given function is that of a semi-circle

Try plotting it on Desmos

[kindred folio]

ohh ok i got it, tysm!

.close

[lone heart]

Channel closed

Closed by @kindred folio

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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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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.

[atomic bobcat]

If so how do I go about doing part c?

Cause I can’t use the values I found in part a or b right? So I’m a bit lost

[lone heart]

@atomic bobcat Has your question been resolved?