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[dense bloom]

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[half garnet]

Hello, i have a question why does i=i' i tried to understand i don't see why. Is it because it's a rhombus. I understand why r=r' but for i, i don't know . Thank you

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[half garnet]

.reopen

[lone heart]

✅

[half garnet]

<@&286206848099549185>

pls

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[half garnet]

.close

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[short thistle]

Hey guys, I have some trouble representing a volume.

[short thistle]

Like I try to project onto z=0, but I get 0<= x^2 + y^2 + 1

which is -1 <= x^2 + y^2 which is impossible

And I know 0 <= z <= 2 cuz x^2 + y^2 <= 1

is the x^2+y^2 <= implicit? we should deduce that by ourselves?

cuz if thats the case, then I think we get a paraboloïde?

but, z can't get to 2

idk, please help 😢

thanks in advance

and does z start at 1 or 0?

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[alpine sable]

Hey I need help with my geomertic project

[alpine sable]

ill send the instructions over

this part

i need

i have almost ten

i need five more

im doing it on the chichen itza

<@&286206848099549185>

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[serene moon]

Hi

can anyone help me on this question

[vale wigeon]

have you made any progress so far?

@serene moon

[serene moon]

I'm not sure if it's right but I guess I'll try finding the line of intersection between the 2 planes first and then find the angle between that line and the first plane

[vale wigeon]

you have described what the problem wants you to do

[serene moon]

yeah I didn't understand that at first

one of the planes is 3d and the other is 2d I think

[vale wigeon]

no, they are all planes in three-dimensional space.

the first one just happens to be parallel to the z axis so the coefficient of z in its equation is zero

[serene moon]

oh ok

um yeah im stuck

idk what to do from here

@vale wigeon

[queen salmon]

Since the first equation has only x and y, you might be better off getting a system of eqations with only those two

[lone heart]

@serene moon Has your question been resolved?

[vale wigeon]

actually you do not need the line itself

only its direction vector, which can be obtained as the cross product of the normals to the two intersecting planes

sorry for the delay - was getting some paperwork done

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[lean plover]

i did this by working out variance for 1

then multiplying by 100

but mark scheme does something completely different, i cant understand what they do, specifically this line:

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@lean plover Has your question been resolved?

[lone heart]

@lean plover Has your question been resolved?

[olive loom]

this line is just

X = number of heads
100-X = number of tails

so sum of all numbers would be

Y = (1)(X)+(-1)(100-X)

[lean plover]

i kinda get that alright, thank you

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[rancid root]

Hi guys, doing simultaneous equations with one linear and one quadratic. I've rearranged the first equation so I can substitute in y. Simplified, to get a quadratic. I have tried to factor using ac, but not found anything that works so far. Is there another way to approach this or have I gone wrong? Thanks

[tall hearth]

you substituted y wrong

[rancid root]

xd

What do I do differently then?

[high rapids]

The first equation is y=5x^2+6x-7 right?

What way did you rearrange this?

[rancid root]

Yep

[tall hearth]

y = 2x + 3
-y = -2x - 3 ≠ -2x + 3

[rancid root]

+7 on both, -y on both

[high rapids]

Ok

But it's more complicated and you are prone to make mistakes

[rancid root]

Yeah fair enough, it's my first crack at doing these with a quadratic in

[high rapids]

I would just plug in the second equation (already rearranged with y as subject) into the first

So y=5x^2+6x-7 becomes 2x+3=5x^2+6x-7

[rancid root]

So 3=5x^2+4x-7?

[high rapids]

Yeah

Keep solving

[rancid root]

Sorry, gotta write this down, terrible at doing this in my head

5x^2+4x-10?

=0

[high rapids]

Yes

[rancid root]

Then ac?

[high rapids]

-50

[rancid root]

Just meant is that what I do

But ig so

[high rapids]

Yes try that

Whenever the AC method doesn't work use quadratic formula

[rancid root]

Oki

[high rapids]

But here it does work

[rancid root]

Omg I'm an idiot, I saw 4 and thought 5

I can't think of anything

[high rapids]

Nah I am the real idiot

It doesn't work

[rancid root]

Oh alr then

[high rapids]

So use the formula

[rancid root]

k, got x=1.069 or x=-1.87

how do i get the factors using those? i was never taught that, only how to solve

[high rapids]

What factors?

[rancid root]

yknow what im mixing up another question

ignore that

what do i do next? substitute the values in to find y?

[high rapids]

Now plug in each of the values of x to one of the original equations (preferably the second because it's simplier) to find their respective values of y

[rancid root]

awesome, thanks!

off to practice that some more. i made that alot more complicated than it needed to be

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[hollow coyote]

I’m genuinely stuck at what should be the first step in this proof. I generally get how to solve it, yet I cannot find the correct theorems to use in a 2 column proof

[lime bobcat]

Can we use CDB is right?

[stone glen]

I don't think we can

How can one prove congruency ? Like in how many ways?

If u have a query, #❓how-to-get-help

[hollow coyote]

I’ve considered using ASA, AAS, SSS and such

[gray isle]

Can we use CDB is right?
you cant use it immediately because it isn't given

[raven dagger]

All 3 sides equal
2 sides and 1 angle equal
Or 2 angles and 1 corresponding side equal

[lime bobcat]

It could have been writen elsewhere

[hollow coyote]

Nothing else was given

[gray isle]

all the required info is given

[raven dagger]

Here you can use the 2 side 1 angle one to prove it

[lime bobcat]

Okay, thanks, @hollow coyote

[gray isle]

you'd start with what's given

in this case that those two small triangles are congruent

[lime bobcat]

You have AD=DB, ED is common and CDB = CDA

[gray isle]

and use corresponding parts theorems to justify congruent sides and angles,
then try continuing from there

[hollow coyote]

ok

[lime bobcat]

CDB=CDA is the same as saying EDA = EDB

so you have it

[hollow coyote]

I essentially got it now

Just need to write it in a 2 column format now

[lime bobcat]

(btw, CDB being right is straight-forward from the fact that CDB=CDA and A, D, B are collinear)

[hollow coyote]

I think I have it handled now that I know where to start

ty

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[turbid cargo]

Is my answer correct? I'm a bit confused by the wording.

I was thinking the answer could also be

30 x 29 x 28 if dates are not repeated

[stone glen]

Looks correct to me

And i agree that the wording in the question is quite strange, although correct

And makes sense

[turbid cargo]

Ok, thanks!

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[hushed mica]

$sin 3\alpha = sin\alpha(1-2sin^2\alpha+sin\alpha-sin^3\alpha)$

[ocean seal]

Night Wølf

[hushed mica]

$sin3\alpha = 3sin\alpha-4sin^3\alpha$

[ocean seal]

Night Wølf

[stone glen]

@hushed mica what are u looking for?

[hushed mica]

i am trying to derive the sin addition formula to triple angle formula of sin

[hushed mica]

and this is where i have reached till now but i think i have done it wrong

[stone glen]

You can derive this using the addition formula

[hushed mica]

i am doing that only but idk where i did it wrong

[lime bobcat]

$\sin(3\alpha)=\sin(\alpha+2\alpha)=\sin(\alpha)\cos(2\alpha)+\cos(\alpha)\sin(2\alpha)=\cdots$

[ocean seal]

Categorist

[hushed mica]

i already did this

but i think i did some miscalculation

[stone glen]

You did it wrong

[lime bobcat]

We can't say if we don't see your intermediate calcualtions.

[stone glen]

I don't understand how u ended up with this

[hushed mica]

hmm lemme try once more

[hushed mica]

same 😂

[slender gull]

Funny as it is supposed to be, this is not helping you lol.

[alpine sable]

This is correct

[slender gull]

Yeah.

[alpine sable]

Like thats it

@slender gull Imma read all day from now on

[slender gull]

Bet!

Read with your partner infact for all it's worth.

[alpine sable]

ahaha yes of course "partner" yes
if I really had a partner I wouldn't read lol

Bro this is a joke

after reading his bio u see

lol

[slender gull]

Yeah lol.

[slender gull]

Or maybe read 24 * 7?

Jesus.

[hushed mica]

ngl but its fun to troll in this way 😂

[stone glen]

That's not what he was asking about, he didn't end up with that

[hushed mica]

<@&268886789983436800>

[slender gull]

Yeah.

[hushed mica]

but done now

i did some miscalculation

<@&268886789983436800>

@junior blade

[slender gull]

Holy shit.

[stone glen]

;(

[hushed mica]

wtf

[stone glen]

What's going on

[hushed mica]

mods are ded

ded

wk

[lime bobcat]

@heavy otter SPAM

[hushed mica]

<@&268886789983436800>

[stone glen]

Amber Heard of Amber Heard?

[hushed mica]

freak they need a moderation bot

[alpine sable]

nice pun

lol

[stone glen]

:3

[hushed mica]

ok i did some miscalc previous time 😂

[slender gull]

Lol of course you did.

[lone heart]

@hushed mica Has your question been resolved?

[lone heart]

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[topaz copper]

I wasn’t here for this unit so I’m really confused on how to do any of this

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@topaz copper Has your question been resolved?

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[weak seal]

Using a crankshaft, a carpenter drills a 2cm radius hole through a ball of wood along a diameter and the radius of the ball is 5cm. It asks to find the volume of the remaining wood

[lone heart]

@weak seal Has your question been resolved?

[weak seal]

yeah that's what I was thinking but like

would we find the half volume first?

oh yeah that should be right

so like

would it be 2pi*x times 4pi times radius^3/3?

wait this is wrong

or could we find the volume of the revolution between 2 and 5 and substract that to the total volume?

I'm not sure to understand fully

[violet ore]

I think it would be the intersections because the hole doesn't exist outside of the sphere

[weak seal]

ok so like I would just substract r from R and do that squared and times pi?

well yeah

mb forgot to write that

got it thank you

[keen pasture]

Idk, if it's right. It may as well be false

Nice, good to know that

[weak seal]

hmm I am so far away from that wth

well I didnt finish it but here's what I had

[keen pasture]

I'm not integrating the cylinder here. It's just the missing parts of the sphere "above" and under the cylinder

[weak seal]

oh yeah

so yeah rn I'm solving this integral

which is taking me a bit cuz it's been a minute since I last used trig sub

[keen pasture]

You don't need to that. The root and the square cancel

[weak seal]

oh.

thank you for letting me know before doing all that

[keen pasture]

You're welcome.

[weak seal]

found 28pi sqrt(21)

which is around 403 thank you both of you

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[vivid canopy]

how do i find ?/1000 = 7.7

[solid gust]

Multiply 1000 both side.

So you get ? alone on the left.

[vivid canopy]

no

to make 7.7

[solid gust]

Huh?

[vivid canopy]

pls help\

im bamboozeled

[solid gust]

I gave you a solution.

[vivid canopy]

idk what

[solid gust]

Try to have the variable alone on left or right side.

[vivid canopy]

what is the variable and wat is alone on left or right side

[solid gust]

Variable is some unknown.

So the variable in the question is ?

[vivid canopy]

yes

[solid gust]

Usually the variable would be x, but other also works.

[vivid canopy]

nooo not x no letters

[solid gust]

Alone on the left or right side means to give the variable a definition.

[vivid canopy]

idk what you mean still

[solid gust]

Like ?=3

So the variable has a value of 3

In the question we’re gonna find the value to ?

Which is the variable, the unknown.

[vivid canopy]

why is it 3

[solid gust]

Example.

[vivid canopy]

no its blank

[solid gust]

A equation is like a beam balance. If you add 10 on the left, so needs the right, to hold the balance.

[vivid canopy]

its not algebra

its just multiplication

[solid gust]

Ye, so you know the way to solve:)

[vivid canopy]

no i dont

[solid gust]

I’m in a rush right now, if you got other questions, other person could explain:)

[vivid canopy]

i still have the same question

that is not the answer im looking for

someone help

[weak seal]

You re looking for ? In the equation ?/1000=7.7

So for example you have an equation 2=2

[vivid canopy]

no

[weak seal]

I’m giving an example

[vivid canopy]

2=2?

[weak seal]

Yeah let’s say you multiply the right side and the left side of the equation by 2

What would be the equation?

[vivid canopy]

i dont think you understand

____ divide by 1000 = 7.7 what is the answer

[weak seal]

Yep exactly

So you need to find the number if you divide it by 1000 it will give you 7.7

[vivid canopy]

yes

[weak seal]

Do you know what the inverse of the division is?

[vivid canopy]

what does inverse mean

[weak seal]

Contrary

[vivid canopy]

what does contrary mean

[weak seal]

Hmm

What is your first language I might be able to translate it

[vivid canopy]

english

plss

i move decimal where???

[weak seal]

Ok so I’m not sure what would be the best way to explain if what I said didn’t work sorry

Right

[vivid canopy]

?

[weak seal]

You move the decimal to the right

[vivid canopy]

idk what to do tho

its not 7.7

and not 0.077

its ok

ill just ask my teacher

[lone heart]

@vivid canopy Has your question been resolved?

[lone heart]

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[wise raptor]

hello, i need help with this

[wise raptor]

i need to find the derivative using first principles

and i have no idea how to deal with the 1/x

[simple swallow]

U can rewrite 1/x as x to the power of -1

[wise raptor]

hold on,

i still don’t get it:/

[simple swallow]

Is first principles where you’ve gotta put it into that long formula thing?

[wise raptor]

yeah

f(x+h)-f(x)/h

[simple swallow]

Ah shit bro I can’t even help every time I’ve done a test on that I’ve sacrificed the marks for it 😂

[wise raptor]

you’re thinking of the power rule

[simple swallow]

Ye I was lol

[wise raptor]

ahhh okay lol, thank you though

[simple swallow]

Nws I’ll take a look for a sec see if I can figure it out

[wise raptor]

do i go to another help place or will someone come here eventually?

[wise raptor]

okayyy thank you

[simple swallow]

Someone shud come here eventually

[wise raptor]

idk what to do from here on though

[alpine sable]

How

Just write 1/(x+h) - 1/x

Find the common denuminator and then just subtract them

It simplifies

[wise raptor]

i forgot that it isn’t a positive anymore

[alpine sable]

Also limits are linear

So you can find the derivative of 2x and 1/x separately

[wise raptor]

thank you, but my tutor wants me to use that specific method

[alpine sable]

You're using that method

You're just separating the limits

Its one of their properties

[wise raptor]

okay, ill do that, then just add them?

[alpine sable]

Ye

[wise raptor]

i can jsut take the denominator down right? so x+h and x go down to h?

idk if that makes sense

[alpine sable]

You multiply up and down

[lone heart]

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[wise raptor]

thank you! i’ll try that

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[mossy zodiac]

hi amigos! can anybody spot the mistake here?

[tacit arch]

can't even tell what is a function of what

or what's a constant

[alpine sable]

Find dL / dV

But V is in the equation

funsies

[mossy zodiac]

tried to write that stuff in at the top

[alpine sable]

There is no mistake

[mossy zodiac]

yuck

my friend on twitch is getting the same answer

mistake = did not capitalize V

thanks amigos, sorry for waste of time

.close

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[flat mica]

Hi, is my math correct?

[clear stump]

yes

[flat mica]

Thanks

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[clear stump]

Well that was easy

I have no idea btw

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[alpine sable]

what do you do to find perimeter and area

[alpine sable]

You subtract, multiply and add

Not in that order potentially

Do you have any idea

naw

[clear stump]

You have angles and length

so pythagoras do tricks

[plush crane]

But it did not give you the angle of the other side

You have 2 90degrees

[alpine sable]

You have 4 90 degrees

[plush crane]

?

[alpine sable]

You have a rectangle

[plush crane]

How can you be sure

[alpine sable]

Trapezoid, you draw a height

The height there is drawn for you

[plush crane]

2 90degrees cannot prove it is a rectangle

which one

[clear stump]

yeah but it is

[lone heart]

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[alpine sable]

🥲

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[fierce prairie]

idk

[clear stump]

0

000

[digital shard]

lmao

is this even possible wtf

[pearl wraith]

yeah 000

lmao

wait

that's a divison sign

[clear stump]

oh shit

looks like a +

[pearl wraith]

if you could put variables, it would work

like AAA² /AA = AA

but numbers idk

[normal ingot]

AA can be written as 10*A+A

[lone heart]

@fierce prairie Has your question been resolved?

[fierce prairie]

? why

oh wait I see it

[normal ingot]

e.g. 23 = 2*10+3

[fierce prairie]

hm

so where can I go with that

[normal ingot]

if x is the black block, the equation we have is x/(10*A+A)=10*A+A

what happens if you simplify that

[fierce prairie]

x = 22A^2?

[normal ingot]

not quite

did you end up getting $\frac{x}{11A}=11A$?

[ocean seal]

lirmirit

[fierce prairie]

uhh no but what happened to the A?

I did x / (11A) = 11A

[normal ingot]

yes, that is the same as what I wrote

[fierce prairie]

wait im blind lmao I saw 11.4

my bad

[normal ingot]

oh lol

[fierce prairie]

okay yea

[normal ingot]

so we should get $x=121A^2$

[ocean seal]

lirmirit

[normal ingot]

can you see how?

[fierce prairie]

okay yeah

[normal ingot]

now we can just choose any value of A which will make x a three digit number

[fierce prairie]

oh so like 1 makes 121?

and then 121/11 = 11

okay i see that now

thanks

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[alpine sable]

Can anyone let me know what T.P. is referring to here?

[alpine sable]

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[fierce prairie]

I tried
99
+11
110

but that makes B and D the same number

so idk

<@&286206848099549185>

[thorn lotus]

88+22=110

Perhaps try that one?

@fierce prairie

[fierce prairie]

ohh yeah

thanks

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[pale jackal]

hey can i get help w/ this question please?

[pale jackal]

i got tripped up on the "on one trip" line

i eliminated everything else as the best i could tho

the answer is B btw, i just want to know how to get there

[fallow grove]

wouldn't the on one trip just be january and seattle?

[digital shard]

its B

@pale jackal

so u have everything fillled out up to the second bullet point

u can also put a check in november, disney land

and an x in november for the other places right?

[pale jackal]

wait wym i dont get it

[pale jackal]

why would i do this

[pale jackal]

why would i do this as well?

[pale jackal]

how would i get to B?

can you guide me through the process of elimination please

[digital shard]

yes

[digital shard]

so put a check in november, disney

because u know they went to disney, and they didnt go in the other 3 months

so that means they HAD TO go in november

essentially, there should only be one check per column because they went to each place only once

also there should only be one check per row because they went to only one place each month

[digital shard]

so thats why u do that

once you do that, you know that on one trip, the city and month had the same letters

u know its not disney and november, so that leaves january+florida, or january + seattle

however, u also know that they went to florida AFTER seattle, so florida cant be january

which means seattle must be january

also, they went to florida after hawaii

that means hawaii must be may and that leaves july for florida

[pale jackal]

ahhhhh i see

i get u i get u

ok so 1 sec

ok got you thank you brother

is there a limit to how many questions i can ask? i have more btw

these questions that are like puzzles are tripping me out

oh wait i got it

its just the letter corresponds with 3 letters ahead

should be D on this one

wait the correct answer is c, how come?

[edgy sail]

its decode so 3 letters behind

[pale jackal]

ohhhh shit i see

i didnt know vocabulary had an effect

an im assuming if its encode.. then its forward?

[edgy sail]

yeah

[pale jackal]

ahh i see ty ty

[pale jackal]

^ btw

dont wanna bother u guys

[edgy sail]

i think ur good

[pale jackal]

solid sounds good

im stumped on this question, if the less than signs apply to every square how can i fit a number next to the 1 if its going to be wrong?

[lone heart]

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[alpine sable]

hello everyone idk if anyone understand french here but i’ve been stuck on this problem for 2 days and idk what to do i’ve tired different steps but my teacher keep saying it’s wrong

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@alpine sable Has your question been resolved?

[lone heart]

@alpine sable Has your question been resolved?

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[weary crag]

the length of the latus rectum of the hyperbola x^2-y^2=64 is

[weary crag]

Got it nvm

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[lost coral]

is this correct?

[wanton junco]

yea

[lost coral]

ok

tysm

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[brazen crest]

i can do the second part but im stuck on the first and the third qns

[karmic rapids]

for the first, compare the bounds, what do you notice?

for the third you need to know that $$\int_a^b f(x)dx = -\int_b^a f(x)dx$$

[ocean seal]

illuminator3

[karmic rapids]

@brazen crest

[brazen crest]

oh wait

both r 3 so the ans will be 0?

[karmic rapids]

yes

[brazen crest]

thanks so much 🙏🙏

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[alpine sable]

I know its physics but it's still math so can someone help me to know to develop this equation ?

[alpine sable]

The answer is this

I dont know how to go from one to the other 🥺

[harsh girder]

$\frac{Q}{x^2}=\frac{4Q}{4x^2} = \frac{4Q}{(x-d)^2}$, so
$$4x^2=(x-d)^2$$

[ocean seal]

秋水

[alpine sable]

Thank you very much ❤❤🙏🙏

And just for this one how do we go from the first one to the second one ? 👉👈

[harsh girder]

first, $x^2+\frac{2}{3} dx -\frac{d^2}{3}=0$

[ocean seal]

秋水

[harsh girder]

then notice $d-\frac{d}{3}=\frac{2}{3}d$ and $d \times(-\frac{d}{3})=-\frac{d^2}{3}$

[ocean seal]

秋水

[harsh girder]

so you can factor it like that

[alpine sable]

Thank you so much you're my hero 🙇‍♂️

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[tropic stump]

hello!

[tropic stump]

yep

how do i prove it?

[normal ingot]

Can you calculate $\log_{3^{1/2}}81$ by hand?

[ocean seal]

lirmirit

[tropic stump]

yes

[normal ingot]

what do you get?

[tropic stump]

8

oh

[normal ingot]

yes

[tropic stump]

thanks

.close

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[summer berry]

f(x) = 1/x (x >= 1)

[summer berry]

how is the range <0, 1]

if x should be bigger than 1 how can it have a 0?

[balmy warren]

x >= 1 yes?

[summer berry]

yes

[balmy warren]

so why don't you plug in values of x greater than or equal to one

what happens to f(x)

remember the range of f(x) is the values it outputs

[summer berry]

ahh I see the bigger the x the lower f is gonna be

so the range is gonna be zero at a certain point?

[balmy warren]

does it ever reach 0?

[summer berry]

but wait even if you divide 1/1000000000 it will never reach zero

[balmy warren]

correct

[summer berry]

yeah..

ahh so < doesnt include zero

[balmy warren]

correct

[summer berry]

and for 1 the minimum value is 1 because 1/1 = 1

[balmy warren]

1 is the max value

[summer berry]

I see thanks a lot!

[balmy warren]

so 0 < f(x) <= 1

yes?

[summer berry]

wait let mee see

yeah f(x) is bigger than 0 but equals to 1 or lower

so yes

am I correct?

[balmy warren]

is this what you are trying to say?

[summer berry]

yeah? am I saying it wrong?

[balmy warren]

no but the notation I have used is if you want to express the range

[summer berry]

ahh okay I understand

thanks again!

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[alpine sable]

Trying to take derivitive of this using the chain rule

[alpine sable]

I have figured out the inner function on the right

But the outer function doesn't seem to be correctly derived

The mathway solver gives this instead

Which seems to suggest that it only consists of e raised to that function and the whole right aide

Right aide

Side*****

.close

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[alpine sable]

I am trying to take derivitive of the function in blue and trying to find jts result

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[alpine sable]

So i use the chain rule for the outer part

[alpine sable]

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[gray ingot]

yes

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[near hollow]

Once you have it in this form where it's quite simple work with it as equations with variables

[normal ingot]

2 zero rows hence 2 free variables

[near hollow]

$x_1+8x_4=-6$

[ocean seal]

Max..

[near hollow]

$x_2=-8$

[ocean seal]

Max..

[near hollow]

Now yes as you've noticed $x_3$ doesn't feature in any of the equations so it is a free variable

[ocean seal]

Max..

[near hollow]

If you solve the two equations above you see that x2 is clearly fixed but what about x1 and x4?

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@acoustic plover Has your question been resolved?

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[thick yoke]

What did you try to do?

[kindred anchor]

this can be rewritten as
$$2^{x}\cdot 3^{2x}-2^{2x}\cdot 3^{x}-2 \cdot 2^{3x}=0$$

[ocean seal]

QuantumBee

[thick yoke]

yea

[kindred anchor]

ooh

i can divide it by 2^x maybe

it is common in every term

[thick yoke]

yea, thats good

[kindred anchor]

$$=3^{2x}-2^{x} \cdot 3^{x}-2\cdot 2^{2x}$$

[ocean seal]

QuantumBee

[thick yoke]

yes

Do you know what to do now?

There is a trick for this type of questions

[kindred anchor]

idk

[thick yoke]

You divide both sides by: $$2^{x} \cdot 3^{x}$$

[ocean seal]

Ariel1300

[thick yoke]

which is the term in the middle

[kindred anchor]

that makes a lot of sense

how did i forget doing this

so, it then becomes
$$\frac{3^{x}}{2^{x}}-2\cdot\frac{2^{x}}{3^{x}}-1=0$$

[ocean seal]

QuantumBee

[thick yoke]

yes

[kindred anchor]

say $t=\frac{3^{x}}{2^{x}}$
\newline then, $t-\frac{2}{t}-1=0$

[ocean seal]

QuantumBee

[thick yoke]

and here you have a normal quadratic equation

[kindred anchor]

$t=2$ or $t=-1$

[ocean seal]

QuantumBee

[kindred anchor]

Taking t=2,
\newline $\frac{3^{x}}{2^{x}}=2$
\newline $3^{x}=2^{x+1}$
\newline $x\ln{3}=(x+1)\ln{2}$

[ocean seal]

QuantumBee

[thick yoke]

you have a mistake in the last line

[kindred anchor]

ok

got it

[thick yoke]

$x\ln{3}=(x+1)\ln{2} \Rightarrow x=\frac{\ln{2}}{\ln{3}-ln{2}}$

[ocean seal]

Ariel1300

[kindred anchor]

yes

just typed it by mistake

thanks

ariel

[thick yoke]

youre welcome

[kindred anchor]

@thick yoke i have one more doubt

[thick yoke]

what is it

[kindred anchor]

@thick yoke

[thick yoke]

So, you can write: $$\log_{3}{m} = \frac{\log_{2}{m}}{\log_{2}{3}}$$

[ocean seal]

Ariel1300

[kindred anchor]

yes

[thick yoke]

and then we get:

$$m^{\log_{3}{2}} + 2^{\frac{\log_{2}{m}}{\log_{2}{3}}} = 16$$

[ocean seal]

Ariel1300

[thick yoke]

$$m^{\log_{3}{2}} + m^{\frac{1}{\log_{2}{3}}} = 16$$

[ocean seal]

Ariel1300

[kindred anchor]

understood

[thick yoke]

$$m^{\log_{3}{2}} + m^{\log_{3}{2}} = 16$$

[ocean seal]

Ariel1300

[thick yoke]

$$2 \cdot m^{\log_{3}{2}}= 16$$

[ocean seal]

Ariel1300

[thick yoke]

and that's it

Here you just divide by two and then take the "log_3(2) th" root

[kindred anchor]

$$m^{\log_{3}{2}}=8$$

[ocean seal]

QuantumBee

[thick yoke]

and you just put this in a calculator (which I don't have near me rn) and get the result

[kindred anchor]

what if i shouldn't use a calculator

[thick yoke]

let me see

[kindred anchor]

i just found it

[thick yoke]

This works

[kindred anchor]

same

got it

[kindred anchor]

what do you use to render this?

[thick yoke]

I use LyX

and not the discord bot

I use it mainly for writing my HW on university

https://www.lyx.org/

[kindred anchor]

oh

ok

thanks again

have a good day!

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[thick yoke]

You too

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[pale jackal]

hi im back 🙂

[pale jackal]

can some1 help me with this please, where do i begin to plug in the info?

im super confused on the possible values - why is it "possible" how can i begin to sort everything out?

[lament glen]

$|A \cup B| = |A| + |B| - |A \cap B|$

[ocean seal]

Doggo

[lament glen]

so for your case a + c - b = 30 - 4

oh wait we don't need this, perhaps

is the part in red = a?

or does it also include the intersection?

[alpine sable]

That's what I understood, personally

[lament glen]

yeah I think it makes more sense that way

since it's kinda written towards the side n all

[lament glen]

then we have a + b + c = 26

[alpine sable]

Kinda weird that there's students in b at all, considering they said students were offered a choice between a concert ticket and a movie ticket

[lament glen]

hm I wonder if there's a trick with that

tbh

now that you point it out

[pale jackal]

im new to this unit, so can you explain to me what this equation means and why we use it in this scenario - if that isnt too much to ask pls?

[lament glen]

ignore that equation

we don't need it here

[alpine sable]

Pretty sure you just need to pick any a, b, and c values that add up to 26 out of the possible values

[pale jackal]

can u explain it to me regardless pls, n ur thought process as to why u wanted to use that at first? just so i can better understand cuz alot of questions in this workbook use that equation

like for example ik this question uses an equation, but idk where to start

[lament glen]

|A| is the amount of elements in a set

let me draw real quick

to explain the rest

[pale jackal]

appreciate it brother

[alpine sable]

This one's pretty easy. To start, suppose you have n distinct elements, how many ways do you have to arrange them?

[lament glen]

let's say we have a venn diagram that looks like this, and we're given |A|, |B| and |A n B|

[alpine sable]

Or maybe we can do it after we're done with the first one if you want

[lament glen]

to find |A u B|, what we do is we first add |A| and |B|

but that included our intersection twice

so we subtract |A n B|

so that it isn't included twice

[alpine sable]

Basically, |A u B| is everything that is either in A or in B, and |A n B| is everything that is in A and in B

[lament glen]

think of it like this, though badly drawn, see how both green and red include the intersection

[alpine sable]

You can remember "u" as meaning "union" and "n" as being a shortening of "and"

[alpine sable]

Modern art

[lament glen]

well I'm drawing it in paint so don't judge

it has some deep meaning that you don't understand anyways

[pale jackal]

ahh i get ya i get ya

[pale jackal]

yeah i see

[alpine sable]

I literally just learned about all of this hah
I quite literally found this way of memorizing it a minute ago

Like literally his equation made me question if I was qualified enough to answer, so I googled what these symbols meant

[lament glen]

the equation isn't really needed here though

[lament glen]

that is assuming a only includes this part

[alpine sable]

Yeah, so, again, literally just find values for a, b, and c out of the possible values that add up to 26

[pale jackal]

ahh i see

but couldn't that be like a whole combination of numbers, looks like it

[lament glen]

the question did say multiple answers

[pale jackal]

or did they make the question so theres ONLY 3 numbers

ohhh

im dumb didnt c that

ty guys

oh but wait

[pale jackal]

can u explain this one to me as well?

does it have to do w/ making a ratio?

[alpine sable]

Ah, this one's easy

Suppose you have a distinct set of n elements, how many ways are there to arrange them?

[pale jackal]

n elements, wym by that?

[alpine sable]

Like, n things

A bunch of things such that there's n of them

y'know

[lament glen]

apples

[alpine sable]

n apples for example

n whatever honestly

[pale jackal]

numbers?

natural numbers

?

or is 'n' just a variable like x

[alpine sable]

It is a variable, but typically when we say "n" we're assuming it's a natural number

n: natural number
k: integer
x: real number
z: complex number

Thing of it like this, that's how most people use them

Anyways, this is unrelated. Suppose you have n distinct things, how many ways are there to arrange them?

[pale jackal]

would it be limited to the number of n distinct things we have...?

[alpine sable]

...What?

[pale jackal]

im dumb sorry 💀

[alpine sable]

Suppose you have 9 balls, in how many ways can you order them?

[pale jackal]

oh

9 times

or wait no

it would be a big number no?

[alpine sable]

Yes

Ever heard of a factorial?

[pale jackal]

does it have an exclamation point?..

or no

[alpine sable]

I supppose they're trying to have you solve it the hard way, with n choose k or something I dunno. It's just 10 times easier with factorials though

[alpine sable]

Yes

9!

9*8*7*6*5*4*3*2*1

Which is indeed a big number

,calc 9!

[ocean seal]

Result:

3.6288e+5

[alpine sable]

Gee das big

[pale jackal]

ohhhh i see

got ya got ya got ya

so when you do x!

you just times that number by how many digits it has

[alpine sable]

Eh? No

You multiply it with every positive integer smaller than it

12! would be 12*11*10*9*8*7*6*5*4*3*2*1

[pale jackal]

yeah thats what i meant, so our question ZYZZYVA, 7654321

[alpine sable]

,calc 7!

[ocean seal]

Result:

5040

[alpine sable]

Oh

I see what you tried to do

I use a \ backslash before every * to make it not do anything

That's why it takes so long to type

Stupid Discord...

[pale jackal]

loool

but so is that correct, is that all there is to factiorials? thats super easy then

[alpine sable]

Yeah, you can think of it like this:
How many places to place the first letter? 7
How many places to place the second letter? 6, one was taken
How many places to place the third letter? 5, two were taken
...

Except, no

Because there' 3 Zs and 2 Ys, and it shouldn't matter what order these appear in

Like, suppose I write down ZZZYYVA, does it matter if I swap the first and the third Z? No, it's still the same word

But that's easily fixable, with a bit more factorials

See, in any arbitrary word, how many ways are there to order those Zs?

@pale jackal Did I break your brain? I can reexplain anything

[pale jackal]

LOL im just digesting this, sorry

so are you giving me an easier way to plug in factorials or, are you telling me my answer was wrong and i have to a bit more work?

[alpine sable]

Well, I'm saying that 7! might look like it works, but it actual overestimates the amount of ways to arrange those letters

See, if you had 7 distinct things, there would be 7! ways to order them

But here, we don't have 7 distinct things, some of those letters appear more than once

[pale jackal]

ohhhhh i see

[alpine sable]

Like, one way to order the letters is ZZZYYVA. In that case, would it matter if I swaped the first and the third Z? No, it doesn't

[pale jackal]

i get ya

[alpine sable]

But our method still counts those as separate because it thinks all Zs are different and all Ys are different

Right?

[pale jackal]

right

with that in mind, how can we make an equation with it?

it says the answer for this question is 420

i dont c how we can get there

[alpine sable]

Just wait, we're getting there

So, in any arbitrary word, how many ways are there to order those Zs?

Like, you have 3 Zs, if they were different, how many ways are there to order them?

[pale jackal]

3!

[alpine sable]

Yep

[pale jackal]

wait i think i see what ur getting at

do we use some addition here?

[alpine sable]

...No?

And same thing for the Ys, how many ways are there to arrange 2 Ys?

[pale jackal]

so 3! + 2!+ 2!

[alpine sable]

Nonono

[pale jackal]

oh my bad i thought i was genius

https://tenor.com/view/llilb-crying-happy-tears-tears-of-joy-gif-11116649

[alpine sable]

Happens

I think I'm genius everyday

But then reality reminds me that I'm just another ape

[pale jackal]

loool fr

so how would we tackle this then?

[pale jackal]

2!

[alpine sable]

Excellent. See, we had 7!, but that overcounted the amount of ways to order the letters because it counted ZZZYYVA multiple times depending on the order of the Zs, because it doesn't know that each Z is the same and not a distinct letter

So, how many times will it count the word ZZZYYVA? Considering there's 3! ways to order the Zs and 2! ways to arrange the Ys?

@pale jackal Did I lose you there?

[pale jackal]

yeah sorry Im trynna understand wym by "how many times will it count the word ZZZYYVA?"

would we take out the z's and the y's and be left with, VA?

[alpine sable]

We previously came to the conclusion that there must be 7! ways to arrange these letters, right?

[pale jackal]

yes

[alpine sable]

Except that supposes that all of the letters are distinct

But, like, the Zs aren't distinct, are they?

If I have a word formed out of these letters and swap the Zs, it still remains the same word, doesn't it?

[pale jackal]

yes

[alpine sable]

So, we're overcounting the words

Take the word YYZZZVA for example, considering there's 2 Ys in it which can be arranged in 2! ways, and that there's 3 Zs which can be arranged in 3! ways, how many times are we counting this word if we just use 7!?

[pale jackal]

oncve

once

[alpine sable]

No

[pale jackal]

;.;

[alpine sable]

Remember, with 7!, we're assuming each letter is distinct

Meaning that, for example, YYZZZVA and that same word with the 2 Ys swapped are counted as distinct

Same goes for if you swap the Zs, it's gonna be counted as different words

But we don't want that

[pale jackal]

yoyo im sorry to cut you off homie, my mom yelling at me i gotta take her to work quickly, im sorry we can continue this later, i appreciate you taking the time to help me out brother

she been yelling at me for like 5 mins but this question boggling my mind 😭

[alpine sable]

Alright, I'll DM you the explanation for when you get back

[pale jackal]

Ty brother fr, much appreciated ❤️

[lone heart]

@pale jackal Has your question been resolved?

[lone heart]

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[vocal tapir]

I'm failing to understand

why reveal the parentheses first before simplifying what's inside?

I'm just following PEMDAS

then I get a completely different answer

(solved correctly following PEMDAS)

if I simplify what's inside the parentheses first

if simplified, what's inside the parentheses becomes 1

[mortal trellis]

then you made a mistake

[vocal tapir]

in the parentheses the numerator becomes (x+1)(x-1)^2

the denominator becomes precisely the same

don't I have to factor it out first?

[mortal trellis]

no. the numerator is x(x+1)-(x-1)=x^2+x-x+1=x^2+1

note that this is only quadratic in x

[mortal trellis]

instead of cubic like this

[lone heart]

@vocal tapir Has your question been resolved?

[vocal tapir]

wdym

[vocal tapir]

oh I see

[mortal trellis]

just a shortcut to note that it can't be the same as yours. without having to multiply it out

[vocal tapir]

ty

.close

[lone heart]

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Closed by @vocal tapir

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[harsh girder]

$$\log_{a^m}(x) = \frac{1}{m} \log_a(x)$$

[ocean seal]

秋水

[sterile crescent]

hm

i did this

it doesnt lead to anything

[alpine sable]

ig

u should convert

everything

to log base

to 2

using change of basis formula

[sterile crescent]

how can i convert .25 to base 2

OH
OHHH

[alpine sable]

$\log_a{b}=\frac{log_m{b}}{log_m{a}}$

[ocean seal]

sqrt(-1) is approx -30

[alpine sable]

here u will use m=2

and everything will boil down nicely