help-26

[high mulch]

Yeah I think that u^2+7 is inside a square root

[stoic dawn]

according to the table yes

then sub back in for u?

[high mulch]

Wolfram says otherwise

Only the u^2+7 should be inside the root

[high mulch]

Missing a plus sign there

[stoic dawn]

then do i just sub back in for u?

u = x+9

[high mulch]

Yeah, after you add the plus sign, replace u with x+9

[stoic dawn]

just subbing back in looks like

that should be correct

[high mulch]

Missing a plus sign in the log at the end

[stoic dawn]

yes again

[high mulch]

Other than that it's good

[stoic dawn]

ok

i'll have to review it

it's just tough applying everything that i've just learned

[high mulch]

Yeah

[stoic dawn]

in the last 3 weeks

just keep practicing i guess

[high mulch]

[stoic dawn]

couple errors

t = u^2+7

and not sure about my final answer

[high mulch]

Hmm, well the first one looks like maybe they didn't want t9 sub in t, so try just u^2+7 for that

[stoic dawn]

final answer looks right

yes

[high mulch]

For the final blank it looks right

[stoic dawn]

that's what i'm thinking as well

[high mulch]

Maybe distribute and simplify everything?

Some software can be weird about answers not being in the right format

[stoic dawn]

true

i'll come back to it

thanks for your help

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[neon iron]

I know how to do the work, but I am stuck trying to find the next few coefficients.

[topaz sinew]

@neon iron Has your question been resolved?

[sterile finch]

The numbering; c0, c2, c4, ...; looks like you should be skipping the sine coefficents, ie. the zeroes.

[neon iron]

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[neon iron]

To use the discriminant, do we have to set the quadratic equal to 0?

[neon iron]

Like is it something we must do?

what if its just x^2+24x+5

and its not equal to 0

[keen venture]

That's gotta be equal to something, in order to be an equation

Like if you have
x² + 24x + 5 = 4

Then subtract 4 from both sides to get:
x² + 24x + 1 = 0

[neon iron]

oh yeah ur right

okay i see ya

thx

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[late quiver]

Takira is setting up for a cat show. The convention center has set aside 350 square yards of space for the show. The fire code says there must be more than 40 square yards of free space for the walkway in the show. Takira is planning for each exhibit to take up 40 square yards of space. How many exhibits can Takira have in the show?

[late quiver]

You and 4 friends are starting a business
selling bagels. The 5 of you open a
bank account strictly for the business.
You would like to have at least $70 in
the account after you buy the start-up
supplies. You all estimate that the startup supplies will cost $25 and decide
that each person must deposit the same
whole-dollar amount of money into the
account. How much money should each
person deposit into the account so that
you have at least $70 after buying start-up
supplies?

. Vondra read that she should have less
than 15 fi sh based upon the size of her
fi sh tank. She already has 4 tetras in her
tank. She would like to add 2 new types of
fi sh to her tank. She would like the same
number of each new type of fi sh. How
many of each new type of fi sh can Vondra
add to her tank?

[topaz sinew]

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@late quiver Has your question been resolved?

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@late quiver Has your question been resolved?

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[ornate heron]

can someone explain how m1g was divided into its components

i don't get it

like i understand how to do it, but how do i know that angle is theta

like i understand this much i think, but how is theta there

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@ornate heron Has your question been resolved?

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[minor shuttle]

.

[ornate heron]

ah i got it thanks!

[ornate heron]

.close

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[potent breach]

One of the diagonals of a rectangle is formed by joining the points (1, 7), and (9, 5). If the other diagonal lies on the line whose equation is 4x – y = c. What is the value of c?

[potent breach]

I got c=14, which I'm told is correct.

[drifting swift]

and i second that

[mellow arrow]

Use the fact that diagonals of a rectangle bisect each other

[potent breach]

But, 4x-y=14 doesn't seem to be the other diagonal

[drifting swift]

nobody said the rectangle was axis-aligned

[potent breach]

I don't get it

[drifting swift]

do you know what i mean when i say axis-aligned

[potent breach]

One of the sides of the rectangle lies on an axis?

[drifting swift]

no

a rectangle is axis-aligned if both pairs of its sides are parallel to the coordinate axes

[potent breach]

Ok

[drifting swift]

anyway

your rectangle isn't stated to be that

in fact this is what it ends up being

[mellow arrow]

How did you get those co-ordinates (1,5) and (9,7)

[potent breach]

I just guessed it to be

[mellow arrow]

That's the reason you're confused

[drifting swift]

and your guess fell flat.

[potent breach]

What are the equations for this plot?

[drifting swift]

you mean how i made it in desmos?

[potent breach]

Yeah

[drifting swift]

polygon( (1,7), (4,2), (9,5), (6,10) )

no i do not have a method for finding those other two vertices

i just intuited them from the grid

[potent breach]

Hmm

How do I prove 3^{200} > 2^{300}?

Do I use logarithm?

[mellow arrow]

Find such powers of 2 and 3 which are really close, like they differ by 1

[potent breach]

2³=8 and 3²=9, 9-8=1

[mellow arrow]

Yess

Now write 3^{200} as power of 9 and 2^{300} as power of 8

[potent breach]

2^{300}=8^{100} and 3^{200}=9^{100}

[mellow arrow]

and clearly 8^100 is smaller than 9^100

[potent breach]

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[warm maple]

\left(x+a\right)^2+6\left(x+a\right)\left(x+b\right)+9\left(x+b\right)^2

[warm maple]

how do you factorise that? I can't remember lmao 💀

[thorny flame]

Toby

[warm maple]

yuh

completely forgot how to factorise that

[valid marsh]

consider X^2 + 2XY + Y^2

[warm maple]

???

[valid marsh]

do you know the formula for the square of a binomial?

[warm maple]

ofc

[valid marsh]

you can use that here

[warm maple]

dawg

nah idk 💀

im forgetting the most basic things bruh

<@&286206848099549185>

[valid marsh]

you can use this identity to help factor it

[valid marsh]

let (x + a) = X and let 3(x + b) = Y

then you can factor it

[warm maple]

why (x+a) as x and 3(x+b) as y

[valid marsh]

because this allows us to use the identity

as for how I spotted this substitution, I noted that the coefficients are 1, 6, 9, and 6 = 2(3) and 9 = 3^2

[warm maple]

if it was say x^2-10x(y+z)+25(y+z)^2

the answer would be (x-5y-5z)^2

im not trying to like expand it im trying to factor it further

[valid marsh]

similar idea here

[warm maple]

what's the idea

elaborate

[drifting swift]

can you restate your question again

[warm maple]

\left(x+a\right)^2+6\left(x+a\right)\left(x+b\right)+9\left(x+b\right)^2

[warm maple]

^

[drifting swift]

if you were asked to factorize $t^2 + 6t + 9$ would you be able to do that

[thorny flame]

Ann

[warm maple]

but what does

(t+3)^2

have to do with

that

[drifting swift]

there are shorter (and frankly, less entitled) ways of saying "yes, but i don't see the connection"

ok, let's go a step further

if you were asked to factorize $s^2 + 6st + 9t^2$ would you be able to do that

[thorny flame]

Ann

[warm maple]

right.. but I see no interrelation

oddly, i'm trying to split it into two

(s+3t)^2

[drifting swift]

ok great

now replace s with (x+a) and t with (x+b)

and watch the factorization of your original polynomial pop right out

[warm maple]

oh

wtf

wait so

why (x+a) and (x+b) though

[valid marsh]

because those appear a lot in the original expression you sent

[warm maple]

ok so how would I factor this

[valid marsh]

did you replace s with (x+a) and t with (x+b) in s^2 + 6st + 9t^2?

[warm maple]

yes

got my exact original answer

[valid marsh]

so do you know how to factor it

[warm maple]

that's pretty cool

BUT WHAT'S THE ANSWER :cr

ill try to input that correlation

[drifting swift]

you yourself said this: $$s^2 + 6st + 9t^2 = (s+3t)^2$$ now replace things as i said

[thorny flame]

Ann

[warm maple]

i got

my oroginal answer

?

do i factor the same way?

[valid marsh]

Yes

[warm maple]

can you js

please

tell me the answer

bc

[valid marsh]

!nosols

[topaz sinew]

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.

[warm maple]

ive spent 26 minutes on this and ive got

limited time

oh

right

[valid marsh]

did you replace s with (x+a) and t with (x+b) in (s + 3t)^2?

that will be the factorization and so it will be the answer

[warm maple]

(x+a+3x+3b)^2

IS THAT IT

[valid marsh]

because that’s how you factored s^2 + 6st + 9t^2

[warm maple]

W

W MANS

THANKS

.CLOSE

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[neon iron]

Topic: "Basic Math and Logarithm"

My work till now ^^^

I think i did a error somewhere

[valid marsh]

maybe first step

[neon iron]

We cant do it like that?

When bases are same?

[valid marsh]

how did you get from u^2 > 9 to u^2 > 4

[neon iron]

Oh , thats one more inequality from domain

Seprate inequations

[valid marsh]

consider log 0.5

you may have forgotten to reverse the sign

[neon iron]

It doesn't matter in that case ig because

Wait

It do matter

Mb

So signs will get reverse now

[valid marsh]

after redoing your work, what did you find?

[neon iron]

Option a is the one who looks very similar to my answer

[valid marsh]

probably a then

[neon iron]

It is A

Indeed

Idk how they got 5½ in their answer

We can get 5½ tho , if we compare both mumbers

I will close the channel now

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[clever citrus]

How do I approach this

[clever citrus]

y = x if x >= 0

y = -x if x < 0

but like how would this help me prove that our function is a subset of [0, infinity)

wait can I rewrite it as $\sqrt{x^2}$ and then ik that $x^2 >= 0$ for all numbers

[thorny flame]

Calc II Victim

[safe burrow]

i mean, you can prove it has no upper bound by pointing out x has no upper bound

and you can show that there's no instance where |x| < 0

[clever citrus]

this is what I was thinkin

[halcyon slate]

this follows directly from the definition right/

[clever citrus]

definition of?

[halcyon slate]

of |x|

[clever citrus]

I remember the prof saying you can rewreite it as that

[halcyon slate]

that looks fine

[clever citrus]

found it

[halcyon slate]

or you can also take cases

[clever citrus]

lfgggggggggg

for cases how would it work

i thought abt cases first but

didnt know how to prove it

[halcyon slate]

for non-negative x, |x| = x

so f would map [0, infinity) to [0, infinity)

[clever citrus]

but what about

negatives

I cant just say |-x| = x right

[halcyon slate]

for x < 0, -x>0 hence |-x| = -x

so it maps (-infinity, 0) to -(-infinity, 0) = (0, infinity)

[clever citrus]

eh i think transformin it was easier

[halcyon slate]

yeah it's more professional

this is a bit hacky

[clever citrus]

alright so now to prove that $y \in [0, \infty) \subseteq \mathbb{R} (f)$ my prof did a different questiong using the quadratic

[thorny flame]

Calc II Victim

[clever citrus]

but how would I do it

I cant transform my equation to a quadratic

I dont know if just reversing my previous work would be appropriate

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[stark hare]

Need help working out the equation

[regal forge]

try to develop an equation

equation should be: number of cars per month = number of months x constant

[stark hare]

So would it be 40m+12

[regal forge]

yes

now substitute in all of the values in the table into your equation

[stark hare]

Ok thanks

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[worldly flicker]

why is r = sen(delta)

[worldly flicker]

probably really stupid question

sorry

i know it's portuguese but u can get context from image and equations

nvm im dumb

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[small crown]

how do i solve this. Does it even have a solution

[sturdy oracle]

Uh can you write it in a neater way

[small crown]

its Rs

r^n x r x r -1 = r^n x r -1 + r^n x r x (r -1)

[odd pagoda]

wdym with "solve"

you can simplify the RHS

[topaz sinew]

@small crown Has your question been resolved?

[small crown]

i am hardstuck on this induktion example atm

but i am also really tired atm i should probably look at it tomorrow

it was this one

[odd pagoda]

just simplify the RHS and you are done

(or you could have combined the fractions)

[small crown]

idk i should practice my skills for those particular calculations

i dunno how to continue here

[odd pagoda]

multiply out the bracket on the right

then two terms cancel

and you are done

that said, manipulating stuff like this is an absolute essential skill that you must have

practice it

[small crown]

r^n x r x (r-1)

the -1 kinda confuses me

on the far right side

would that mean r^n+2 - r^n+1?

i solved it

@odd pagodasry for that it's just the fact that i am really tired rn

i think i would have solved it myself under normal circumstances

well gn ^^

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[sinful vault]

In what cases are pade approximants better than taylor series and vice versa?

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[trail goblet]

What's the role of n=2 here?

[valid marsh]

That is the size of the sample

[trail goblet]

like 4^n?

[valid marsh]

you are choosing 2 things

that’s what “n = 2” means

[topaz sinew]

@trail goblet Has your question been resolved?

[trail goblet]

what are those things?

sorry English isnt my first la language so I'm having troubles understanding it

[valid marsh]

the things in the sample(which are 0,0, and 0,2, and so on)

[trail goblet]

oh

So if I have n=3

I'll have like 1,2,3

?

[valid marsh]

I think so

by the way why is “algebraic geometry” in your roles

[trail goblet]

what

oh idk

must've been an accident

thank you

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[neon iron]

arithmetic sequence

[keen matrix]

an arithmetic sequence is defined as one beginning term then the following terms have a common difference added to them

[potent breach]

https://www.youtube.com/watch?v=XZJdyPkCxuE

[neon iron]

Why do we use it?

What is its use in our daily life?

[neon iron]

its the most basic series, you wont understand further thinks like AGP, GP etc without AP

it makes it easy to compute weird series

Well, but I'm talking about what practical applications it has?

[neon iron]

you can just google that

moreover, maths applied on real life is physics

series are highly used in physics

like taylor series for approximations

[neon iron]

applied physics is chemistry, applied chemistry is biology, its all maths

[sinful vault]

gp is used in stuff like emi calculations and recurring deposity

[neon iron]

it always has been maths

Why do we have to learn the arithmetic sequence?

[sinful vault]

so you can understand how to calculate emi calcuations and rd's

further, they act as a stepping stone to more advanced math

[neon iron]

yep

[sinful vault]

plus maths has applications of random things in random places, for example, the binomial theorem was used to find the value of pi, both are totally unrelated but still used together. The binomial thereom itself has no "real life" meaning, but it does when you apply it in the right places

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@neon iron Has your question been resolved?

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[true drum]

can someone please help me solve this problem. these are the only ones that i don’t understand

[true drum]

this is what i got so far when i was getting help yesterday

[neon iron]

you know what cos is in a right angled triangle right?

[true drum]

i know that it’s y/x

and yes i learned that

so why exactly do i solve for Sin if it only shows in the problem Cos= -2/3 and tan <0?

oh okay got you

i think

so did i set it up right when i wrote it down then?

.close

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[shadow ember]

In how many ways can we put 10 boys and 10 girls in a row so that same-sex people don't get together?

[shadow ember]

there is an answer in the book, but I don't have a clue on how to get there

it doesn't make sense to me

don't know where to start

[quartz geyser]

2..?

[frozen path]

.close

[topaz sinew]

Channel closed

Closed by @frozen path

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[long stirrup]

how many ways if there's 1 girl and 2 boys?

.reopen

[frozen path]

.reopen

[topaz sinew]

✅

[frozen path]

Ohhh I didn’t know I could do it

That’s cool

[quartz geyser]

1 ig.

[long stirrup]

no there's 2

because you can swap the boys

[quartz geyser]

Ooohoo

[long stirrup]

so that's it, now do 2 boys and 2 girls

and then 10 each

[shadow ember]

2 * 2!?

[quartz geyser]

Bro, use permutation.
(10!)(10!)

[shadow ember]

I got this

but why the 2 in 2*10!10!

[quartz geyser]

Cause u can interchange the places of boy with girl and girl with bow.

[shadow ember]

oh got it now

thanks!

.close

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Channel closed

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[frozen path]

.reopen

[topaz sinew]

✅

[frozen path]

.close

[topaz sinew]

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[neon iron]

I cant solv eit

[prisma mesa]

solve what?

[topaz sinew]

@neon iron Has your question been resolved?

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[neon iron]

Hi , could someone teach me how to do this question? I don’t know where to start!

[gusty bane]

how can you find the intersection of f(x) and g(x)?

[neon iron]

I can solve by substitution!

[neon venture]

?

[gusty bane]

what's another way

[neon iron]

But how do I determine the value of k??

[neon iron]

Elimination?

[gusty bane]

that's what im getting to

no

[neon iron]

Graphing

[gusty bane]

what you're describing works for systems of linear equations typically

let's write it as y = x^2 - 6x + 14

and y = -x^2 - 20x + k

how can we find the intersection/s

[neon iron]

Uhhhhh I just know substitution elimination and graphing. I can’t think of any other ways to find the intersections?

[gusty bane]

you set them equal to each other

[neon iron]

?

[gusty bane]

x^2 - 6x + 14 = -x^2 + 20x + k

[neon iron]

Isn’t that just substitution

[neon venture]

elimination and substitution are things for system of equations, we dont have a system of equation here

[neon venture]

No

[gusty bane]

in a way, yes

we can just proceed

[neon venture]

Nah

[gusty bane]

yes

you substitute y for f(x) or g(x)

it's a roundabout explanation but let's not get caught up in technicalities

[neon venture]

Thats just saying f(x) = g(x) and its not substitution

[gusty bane]

@neon iron

so we have the equation up to

[neon iron]

Yes

[gusty bane]

x^2 - 6x + 14 = -x^2 + 20x + k

would you know how to solve this?

[neon iron]

Get everything to one side?

[gusty bane]

yes

so

i'm going to speak more generally here

[neon iron]

I will move everything to the left

[gusty bane]

before you do

let me explain the process

[neon iron]

Ok

[gusty bane]

you'll get something along the lines of

ax^2 + bx + c = 0

[neon iron]

Yes

[gusty bane]

where c is something like (k + 5)

that 5 is arbitrary, it could be another number

[gusty bane]

so you will get something like this

how do you determine the number of solutions?

[neon iron]

Discrim?

B^2-4ac?

[gusty bane]

yes

and what condition on the discriminant

gives you exactly 1 solution?

[neon iron]

It must = 0?

[gusty bane]

yes

and that will give you an equation for k

do you think you can tackle it from here?

[neon iron]

Uh

Could you please continue to walk me through?

[gusty bane]

i think it would be beneficial for you to try it yourself

move the terms over

and find the discriminant, then set it to 0

try it out and let me know if you get stuck

[neon iron]

Okay.

@gusty bane im stuck

[gusty bane]

what do you have so far

[gusty bane]

you don't need the square root

we just want the discriminant b^2 - 4ac

[neon iron]

Oh

[gusty bane]

so what is the discriminant?

[neon iron]

84-8k?

[gusty bane]

yes, assuming your arithmetic was right

now

as we said

we want a single solution

what must the discriminant be equal to

[neon iron]

Uh 0?

[gusty bane]

yes

so set 84 - 8k = 0

and solve for k

[neon iron]

Okay.

@gusty bane I got k = 21/2

[gusty bane]

yes

[neon iron]

The answr in the back is **-**21/2 tho

[gusty bane]

there's an error in your picture

what is -8(14-k)

[neon iron]

Oops

-112 + 8k

[gusty bane]

fix that and you'll get -21/2

[neon iron]

I am making lots of silly mistakes today for some reason

I have a test tomorrow

[gusty bane]

better today than tomorrow

also focus on getting the logic down so you can do the problems quickly, then check your answers when you're done

[neon iron]

Yeah, even yesterday I was making silly mistakes I don’t know why

Okay.

[gusty bane]

good luck on your exam

[neon iron]

Thanks!

@gusty bane one last question. What if this problem had also asked to find the point of intersection?

[gusty bane]

you plug the k back in

then solve the quadratic any way you know how (e.g. factoring, or quadratic formula)

that gives you the x value

[neon iron]

So 84+8(-21/2)

[gusty bane]

and you can plug that x value into either function to get the y value

[gusty bane]

that's the discriminant

[neon iron]

Oh I plug the k value back into the original

[gusty bane]

so you'd do 2x^2 + 14x + 14 - (-21/2)

yes

and solve this to find the x value

let's say the x value is 10

then you'd plug 10 back into f(x) or g(x) to get the y value

[neon iron]

So I could just plug in there?

[gusty bane]

yes

and solve

[neon iron]

Got it

Thanks

@gusty bane can you teach me this question?

[gusty bane]

what is the height of the rocket?

at a given time t

as in, what expression gives you the height of the rocket

[neon iron]

H(t)=-16t^2+96t

[gusty bane]

yes

so h(t) is the height

so if we want to find the time when the height is 80 feet

we would want h(t) = 80

right?

[neon iron]

Yes

[gusty bane]

that's a quadratic equation you can solve

-16t^2 + 96t = 80

[neon iron]

Complete the square?

[gusty bane]

the 100 feet question is the same thing

sure

or just use the quadratic formula

just solve the quadratic any way you know how

[neon iron]

@gusty bane do I bring the 80 over to the left?

[gusty bane]

yes

[neon iron]

But how do I find the time

Oh isolate t?

My answers are both wrong for some reason

@gusty bane

[gusty bane]

that's a lot of work i honestly don't want to read

have you tried just using the quadratic formula

[neon iron]

I will try now.

@gusty bane but how do I know which of the 2 x values are correct?

[gusty bane]

both are

[neon iron]

Uh

[gusty bane]

oh

the question asked “at what time will it first get to …”

so you take the earlier obed

ones*

[neon iron]

Aren’t I supposed to plug them back in and see which works?

[gusty bane]

they both work

that’s why they are solutions

[neon iron]

There are 4 in total

[gusty bane]

yes

you take the earlier for each pair

reread the question

it says “when does it first reach x height”

[neon iron]

What if I had the 5 in the earlier position

Then wouldn’t I be wrong

[gusty bane]

is t = 5 earlier than t = 1?

i'm saying earlier in terms of time

[neon iron]

Oh

Oh ok

[gusty bane]

so the smaller value

[neon iron]

Oh kk

[topaz sinew]

@neon iron Has your question been resolved?

Channel closed

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[vapid lava]

help please

[mellow arrow]

the two triangles are similar

Use property of two similar triangles

[vapid lava]

still don’t get it

[vapid lava]

still confused

[mellow arrow]

$\frac{3}{6}=\frac{5}{x}$

[thorny flame]

B-eard

[valid marsh]

what do you not get?

[vapid lava]

I solved it all good

[topaz sinew]

@vapid lava Has your question been resolved?

[topaz sinew]

Channel closed

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[drifting swift]

not funny

[uneven halo]

Idk how to solve this

[uneven halo]

It's not a bad joke or something

Someone just sent me this

And idk how to solve it

[drifting swift]

then whoever sent you this played a cruel joke on you

https://www.quora.com/a-b-c-b-a-c-c-a-b-4-What-will-be-values-of-a-b-c

[long stirrup]

we've a command

!elliptic curve meme

[topaz sinew]

🍎 = 154476802108746166441951315019919837485664325669565431700026634898253202035277999
🍌 = 36875131794129999827197811565225474825492979968971970996283137471637224634055579
🍍 = 4373612677928697257861252602371390152816537558161613618621437993378423467772036

[neon venture]

Lmaooo

[topaz sinew]

@uneven halo Has your question been resolved?

[uneven halo]

Bruh

Thank u guys

[topaz sinew]

Channel closed

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[visual merlin]

Hello, all. I am working a problem that asks me to determine the concavity of the function y = x^2 + x + 1. Since y'' = 2, one may mistakenly think that y has no concavity, since y'' = 2 is constant. However, upon checking my textbook's answer key, I discovered that y is concave up for all x; this makes sense, since y goes from decreasing to increasing at x = -1/2. Nevertheless, to ensure my problem-solving strategy is correct, can anyone verify that this – checking where y is increasing and decreasing (the first derivative test) – is the correct way to determine concavity when dealing with a second derivative function that is a constant? Thank you in advance.

[wintry phoenix]

When we talk about the "concavity" of a function, we're basically talking about whether it's shaped like a "U" (which we call "concave up") or an "n" (which we call "concave down").

If the second derivative is positive over an interval, the function is concave up over that interval. If the second derivative is negative, the function is concave down.

For the function y=x^2+x+1, it turns out that it's always shaped like a "U", no matter what value of
x we plug in. This is what we mean when we say the function is "concave up over the entire real line".

[visual merlin]

So, whether the second derivative is a constant or a polynomial (whether there is more than one zero of the second derivative) doesn't matter; if it is positive it is concave up, and if it is negative it is concave down. Do I have that right?

[neon venture]

yea

[wintry phoenix]

That's correct.

[visual merlin]

Thank you.

.close

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[ionic oar]

$\sum_{n = 1}^{\infty} \arctan \frac{2^{n - 1}}{1 + 2^{2n - 1}}$

[thorny flame]

NEONPerseus

[pseudo jetty]

!show

[topaz sinew]

Show your work, and if possible, explain where you are stuck.

@ionic oar Has your question been resolved?

[ionic oar]

I don't know how to begin

[pseudo jetty]

well first off what do you think of when you see
[ \map \arctan {\f {\text{blah}} {1 + \text{blah}}} ]

[ionic oar]

I was trying to split it up with $\arctan x + \arctan y = \arctan \frac{x + y}{1 - xy}$

[thorny flame]

NEONPerseus

[pseudo jetty]

yeah okay so like

just solve the system lol

[
-xy = 2^{2n - 1}, \quad x + y = 2^{n - 1}
]

[ionic oar]

Huh

That's it?

Let me give that a shot

[pseudo jetty]

well the solution should be "obvious"

given that you're trying to sum this thing

[ionic oar]

Is this a geometric progression in disguise

[pseudo jetty]

what?

[ionic oar]

This system is proving increasingly annoying to solve

Do I make a quadratic

Pain

[pseudo jetty]

let me put it another way

[
\map \arctan x - \map \arctan y = \map \arctan {\f {x - y} {1 + xy}}
]

[pseudo jetty]

[
x - y = 2^{n - 1}, \quad xy = 2^{2n - 1}
]

[ionic oar]

Yeah that's what I was doing

Got it nvm

[pseudo jetty]

[ionic oar]

Thanks for your help @pseudo jetty

It's pi/4

[pseudo jetty]

wot

okay

[ionic oar]

Mb

[pseudo jetty]

there we go

but like

surely you can see that [ 2^{2n - 1} = 2^n 2^{n - 1}, \quad 2^n - 2^{n - 1} = 2^{n - 1} ]

[pseudo jetty]

without having to solve the whole quadratic

[ionic oar]

Yeah it clicked later

I didn't solve the quadratic

[pseudo jetty]

okay good

[ionic oar]

Staring at stuff does wonders occasionally

Thanks again

.close

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[pseudo jetty]

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[plush wind]

Hey, today I faced this puzzle and I wonder if there is any mathematical way to solve it?

Basically, u need to connect each tile so it has n (number on the tile) connections

[warm mesa]

id start with doing stuff like

the 3 in the top corner has to have at least 1 bond going both south and west

j fiddle around with it

what game is this

[plush wind]

Adventure Escape Mysteries

[halcyon spade]

solve this?

[halcyon spade]

it looks fun lets have a go

[topaz sinew]

@plush wind Has your question been resolved?

[topaz sinew]

@plush wind Has your question been resolved?

Channel closed

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[neon iron]

yo

[neon iron]

can someone help me out

I don't know where to begin to be honest

[summer dock]

i would draw it out

[neon iron]

tbh I don't really understand what pq is

like is it p + q

p - q

q- p

[summer dock]

it’s a line from P to Q

i guess it’s q-p

but i don’t think it’s a vector

it’s just a line

[summer dock]

ok

R is somewhere on that line

[neon iron]

aren't there infinite possible values

[summer dock]

so what values of PR/PQ can exist

yes

[neon iron]

ah

it was that simple

thanks

[summer dock]

there is a bound

[neon iron]

what does that mean

if I write down k = R+

all real numbers

would that be wrong

[summer dock]

yes

because R must be in between P and Q

[neon iron]

right but for k

[summer dock]

PR <=PQ

and PR >= 0

[neon iron]

right so k is any number greater than 0

ah

so all positive real numbers

[summer dock]

and less than ||1||

[neon iron]

ah gotcha

thanks

how would I use that knowledge to solve

[summer dock]

hmm

so

R is p + k(q-p)

go back to the drawing

and add arrows

[neon iron]

what do I add the arrows for

[summer dock]

why not

it doesn’t matter

[neon iron]

like from where

[summer dock]

oh

from the origin for p and q

and from P for (q-p)

[summer dock]

do you understand this

and how i got this

[neon iron]

i don't

[summer dock]

so p just takes you to P

[neon iron]

ah wait I see

you go to P

[summer dock]

and because k is PR/PQ, and PQ = q-p, PR is k(PQ) = k(q-p)

[neon iron]

and then you subtract like the ratio to decrease the height

[summer dock]

yes

[neon iron]

and then you add the (ratio * q) for the width

[summer dock]

not q

that vector is not q

it is (q-p) remember

[neon iron]

ah yes

okay

got that

[summer dock]

then you can expand the bracket

[neon iron]

ok so you get p + kq - kp

[summer dock]

then you can factorise out p to get a value of λ

[neon iron]

and this is equal to p(1-k) + kq = lambda(p) + q(u)

[neon iron]

yeah gotcha

[summer dock]

yes

so

[neon iron]

1-k = lambda

[summer dock]

yes

[neon iron]

k = q

[summer dock]

yes

[neon iron]

k + 1-k

= 1

[summer dock]

well done

[neon iron]

thank you man

really helpful

enjoy your day

.close

[topaz sinew]

Channel closed

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[summer dock]

µ but it’s the same proof

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[mellow kiln]

did i do this right?

oh wait

315 instead of 305

[green latch]

Yupp

9-305/x² is wrong also (9x²+315)/x

[mellow kiln]

wht

[mellow kiln]

how

[thorny flame]

[code{RED}]

[green latch]

$9x+315/x
\newline =x.9x/x+315/x
\newline =9x²/x+315/x
\newline =(9x²+315x)/x$

[thorny flame]

𝓐𝓡𝓝𝓐𝓑 𝓟𝓐𝓛

[summer dock]

$105 = \pi r^2h$

[mellow kiln]

I dont understand

[thorny flame]

[code{RED}]

[mellow kiln]

yes I agree

[summer dock]

then $h = \frac{105}{\pi r^2}$

[thorny flame]

[code{RED}]

[summer dock]

oh that’s weird

anyway

$P = 9\pi r^2 + 6\cdot 105/r$

[mellow kiln]

why 6/r

[summer dock]

because

nvm

[thorny flame]

[code{RED}]

[mellow kiln]

where is 6 coming from

[summer dock]

from the price of 3

times 2pirh for curved surface area

and h is 105/pir^2

then you minimise

[mellow kiln]

ohhh okay

yes

[summer dock]

$0 = 18\pi r -630/r^2$

[thorny flame]

[code{RED}]

[summer dock]

so $630/r^2 = 18\pi r$

[thorny flame]

[code{RED}]

[mellow kiln]

so

r = 2.23

[summer dock]

,calc 630/18

[thorny flame]

Result:

35

[summer dock]

$r = \sqrt[3]{35/\pi}$

[thorny flame]

[code{RED}]

[mellow kiln]

and h = 6.72

[summer dock]

then sub that into this

,w \sqrt[3]{35/\pi}

[thorny flame]

Results provided by WolframAlpha

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

Requested by [code{RED}]#8104

[mellow kiln]

okay perfect thanks boss

[summer dock]

,w \frac{105}{\pi (\sqrt[3]{35/\pi} ^2}

[thorny flame]

Results provided by WolframAlpha

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

Requested by [code{RED}]#8104

[summer dock]

huh

[summer dock]

that’s not what i got

,w \frac{105}{\pi 2.23^2}

[thorny flame]

Results provided by WolframAlpha

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

Requested by [code{RED}]#8104

[summer dock]

oh

[mellow kiln]

well it is the answer im pretty sure

[summer dock]

it’s just buggin

[mellow kiln]

yupp

alrightyyy thank yu sir

[summer dock]

don’t forget to .close

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@mellow kiln Has your question been resolved?

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[lethal sphinx]

I'm stuck on part a. Can't work out what the loci represent

[lethal sphinx]

I assume c1 is a perpendicular bisector but I'm not sure between what

[topaz sinew]

@lethal sphinx Has your question been resolved?

[lethal sphinx]

Anyone?

<@&286206848099549185>

[topaz sinew]

@lethal sphinx Has your question been resolved?

[lethal sphinx]

No

[valid marsh]

You can try drawing the graphs, or calculating some points on them

[lethal sphinx]

I have

But I have no idea what c1 looks like

[valid marsh]

you can compare c1 with the general equation for a perpendicular bisector

can you rewrite c1 into the form of a perp bisector equation?

[lethal sphinx]

No

[valid marsh]

it’s possible, and if you do it, you’ll know what c1 is a perp bisector between

[lethal sphinx]

But how do I do it

[valid marsh]

What is the general form of a perpendicular bisector?

It should be a formula you can find in your textbook or by google

[lethal sphinx]

The line where all points are equidistant from both lines?

Is it meant to be an equation

[valid marsh]

yes, the general equation of a perp bisector should look something like y = mx + c or |z| = r or y^2 = ax

[lethal sphinx]

So do I just divide the right line by 2

Will that give me it

[valid marsh]

I don’t understand

[lethal sphinx]

Neither tbh

The answer says the line giving the perp bisector is 2d^2+18

Which is just the line on the right divided by 2

[topaz sinew]

@lethal sphinx Has your question been resolved?

[topaz sinew]

@lethal sphinx Has your question been resolved?

[topaz sinew]

@lethal sphinx Has your question been resolved?

[topaz sinew]

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[arctic moon]

Can someone help me understand why I would use factorization here vs. using the limit theorems?

[arctic moon]

Is it because the testing the denominator with the limit as it approaches 14 comes up as 0?

Or is it because the numerator is in quadratic form?

[lyric gale]

t-14 gets cancelled if u factorise

so limit is easier to compute

[arctic moon]

The homework factorizes it

I'm not sure why

Here's an example

[radiant tapir]

you factorize it to find common terms in the numerator and denominator

see how it goes from this to this? They eliminated the (t-2) in the num and dom

[arctic moon]

Yeah

So wait

Is it just picking which is easier?

Between solving it with factorization or limits

they both give the same answer if I remember correctly, no?

[radiant tapir]

well plug in t = 14 into your equation without factorizing. You'll see that you get something divided by zero. So the first trick usually to try with rational limits is to see if anything can be factorized and eliminated

[arctic moon]

That was my original question yeah

So basically the test is

plug in the limit into the denominator

see if it comes out as 0

if it does, then you want to try factorizing first

rational being that there are no imaginary numbers or square roots, right?

[radiant tapir]

yep. It depends on the problem but specifically for rational limits (aka f(x)/g(x) ) then factorizing is a natural first step

rational means a/b

so like let f(x) = x^2 + 3x - 10 and g(x) = x^2 - 4. Then h(x) = f(x)/g(x) = (x^2 + 3x - 10)/(x^2 - 4)

a rational function is that where you have a function in the numerator and a function in the denominator

like how 2/3 is a rational number, but this time you're playing with functions

[arctic moon]

That makes sense

So it would have to be a function in the num/denom

when would that not be the case?

If you had multiple functions?

Such as with f(x+h)-f(x)/h and stuff

or when its basically two clearly defined functions

[radiant tapir]

not sure I understand your question but simply put if you have a function over a function then it's a rational function. Could be polynomial, constant, roots, trig, etc. For example, tan(x) is a rational trig function since tan(x) = sin(x)/cos(x)

[arctic moon]

Yeah

also so...

I think I'm going crazy but

t^2+3t-304

[radiant tapir]

Technically something like f(x) = 1/3 is also a rational function.

[arctic moon]

factorizing this, the factors are 19 and 16

?

[radiant tapir]

yes correct

[arctic moon]

So for some reason

my brain can't figure out how to make 19 and 16 equal +3 when added and -304 when multiplied

It seems like an obvious question

but I think I've been doing math for too long

oh

yeah

I'm just dumb

its -16 and positive 19

[radiant tapir]

there ay go!

[arctic moon]

I was trying to subtract to get 3

I literally just said the word "added"

So the second part was tripping me up a bit

t^2-256 factoring

it would be t^2+1-256 no?

But like, finding what would be a factor of +1 and -256...

[radiant tapir]

ok let's take a step back

is 256 a perfect square?

[arctic moon]

Yes

I can't remember of what

I think 8

[radiant tapir]

nah 8^2 = 64

it's 16

[arctic moon]

Oh perfect square

dur

[radiant tapir]

so then you have t^2 - 16^2 and then you have a diff of squares

which means you can factor it like so : (t - 16)(t + 16)

[arctic moon]

Oh..

Because its 0

not 1

if there was an x it would be 1

[radiant tapir]

well if you foil it out then you'll see as well
(t-16)(t+16) = t^2 +16t - 16t -256 = t^2 - 256

[arctic moon]

Yeah

Because factorization is just basically un-foiling a quadratic.

I'm sure there is a more technical term for it but

[radiant tapir]

so this is how I would handle rational polynomial functions:

1. see if you can factor the denominator somehow
2. see if anything in the numerator can be factored with one of the denominator factors
3. eliminate

[arctic moon]

so when would you not want to factor?

when the denominator when tested with the limit wouldn't equal 0?

Then use limit theorems

or

The one theorem where you set the function to the limit and solve

Limit of a rational function

[radiant tapir]

well I don't think there's a "don't factor" with these. You just check to see if it is factorable in the first place. Often it's easier to see things with polynomials if you can factor them

so I would say try it out and see. Sometimes you get something that's not factorable in the real number system such as x^2 + 1