k
#help-36
hybrid heath
just plug in x=6.2 or x=-6.2
hybrid heath
You basically are trying to solve $xy=-30$ and $x+y=-1$
soft zealotBOT
SWR
deft fjord
ik
simple flame
Isolate an equation for either y or x from either equation and plug that into the other equation. You should get a quadratic which can be solved.
Identify the possible values for both variables and then check whether they are solutions to the original equations.
thorn stump
If two coins are tossed 5 times, what is the chance that there will be 5 heads and 5 tails?
final saddleBOT
If two coins are tossed 5 times, what is the chance that there will be 5 heads a...
thorn stump
dont really get how do i proceed further
do i take indicidual cases like
HH
TT
HT
TH
mint orbit
i think theyre saying total
thorn stump
yeah i understand that
thorn stump
no i mean
HT
as in heads in first coin
tails in second
and TH as in
tails in first
heads in second
theyre different right?
latent fractal
first of all, there's no difference between throwing 2 coins 5 times, 5 coins 2 times, or 1 coin 10 times
thorn stump
yeah thats what i thought
latent fractal
they're all independent either way
mint orbit
i dont think we are differentiating the coins, only the tosses
personally
im usually wrong tho
thorn stump
but i cant seem to figure out how do i count the favourable cases
latent fractal
how many ways can you toss a coin 10 times and end up with 5 heads and 5 tails?
mint orbit
so id take the ht ht ht ht ht to be same as any intoss permutation
latent fractal
how many outcomes are there if you toss a coin 10 times?
thorn stump
wait wait
mint orbit
this will lead to a smaller number that invariance is saying tho
latent fractal
so id take the ht ht ht ht ht to be same as any intoss permutation
I'm not sure I understand your interpretation. Are you thinking a head-tail pair for each of the five tosses?
mint orbit
im saying that the coins are indistinguishible within a toss
so for fewer tosses
ht hh is the same as th hh
but not hh ht
latent fractal
but not hh ht
but all we care about is the number of heads and number of tails
all three of those must be the same
mint orbit
so should i keep it open for a while?
you can do whatever youd like
thorn stump
nah yall are discussing thats why
mint orbit
but all we care about is the number of heads and number of tails
its just up to the interpretation of the language i think
its implied to me that the number of coins matte
latent fractal
also btw i got my answer
if you're satisfied, you can calose
thorn stump
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paper kite
how do I multiply those two radicals?
final saddleBOT
how do I multiply those two radicals?
vital crag
$\sqrt{a}\sqrt{b} = \sqrt{ab}$
soft zealotBOT
riemann
paper kite
I see
something like this?
isn't it a difference of squares?
or am I missing something
how did we end up at (x - 2)(x + 1)
oh I see
you just factor this right
vital crag
isn't it a difference of squares?
doesn't look like it no
vital crag
$a^2 - b^2$ is a difference of squares what is b in your expression?
soft zealotBOT
riemann
vital crag
this says sqrt(x-2)
vital crag
is x-2 even right
vital crag
your handwriting isn't clear enough
paper kite
I wrote everything wrong
vital crag
pen + paper > mouse
paper kite
yea but
how would I share it here
now i'm pretty sure
it is a difference of squares
right?
marsh temple
You can take a photo with your phone no?
paper kite
it's not as fast
marsh temple
Legible beats fast
vital crag
why do things slow and right when you can do it fast and wrong
marsh temple
Making our lives easier helps us help you 😌
paper kite
right
if we use difference of squares
you get something like
I meant
still unsure how do you factor this
vital crag
you have $x -2\sqrt{...} = 4$. solve for the root and square both sides again
vital crag
oh i missed the 2
soft zealotBOT
riemann
paper kite
**riemann**
so do I get all the x's on the other side?
not sure what you mean by this
vital crag
root = $\sqrt{...}$
soft zealotBOT
riemann
paper kite
so we essentially not care about simplifying yet
1 sec
I just square everything now right
vital crag
you should be using this too
paper kite
the thing is
how is this equal to
isn't a = x and b = sqrt(x + 2)
therefore it's x^2 - x + 2
oh I see
it's - (x + 2)
alright I got it
thanks for the help 👍
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uncut birch
Prove or disprove by finding a counterexample: If a | b and a ∤ x, then 𝑏 ∤ x
uncut birch
The first thing I did was take the contrapositive of this statement, which is:
If b | x, then a ∤ b or a | x
Next, to run a proof via contradiction, I take the inverse of the conclusion:
If b | x, then a | b and a ∤ x
If b | x, then b*n = x (where n is an integer)
If a | b, then a*m = b (where m is an integer)
a | b can be rewritten as a*m*n = b*n = x
Which contradicts a ∤ x
So by proof via contradiction, If a | b and a ∤ x, then 𝑏 ∤ x
Except I've heard there are counterexamples, so my proof is wrong
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uncut birch
<@&286206848099549185>
flint parrot
What is the issue dawg
uncut birch
The proof is wrong because there are coutnerexamples
Actually, are there counterexamples
I thought I found earlier but I guess not
But is there anythin else I'm missing
Eh nevermind
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last marlin
If you pull 4 cards without replacement from a deck, what is the probability of pulling out a 2 of any suit, a 3 of any suit, a 4 of any suit, and a 5 of any suit (in any order)?,
final saddleBOT
If you pull 4 cards without replacement from a deck, what is the probability of ...
last marlin
why cant i just do 4!/(52C4)
sturdy cypress
what's 4!
oh
assigning suit to a number
well the suits can repeat
you can do 4^4 / 52c4
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full marsh
need help first finding slope-intercept form, then converting it into standard and/or general
final saddleBOT
need help first finding slope-intercept form, then converting it into standard a...
full marsh
i have the slope and a point for this one, but for some of the other ones i just have two points or have to find a line that's parallel or perpendicular to another line
currently suffering in equation hell
please ping if you answer
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full marsh
<@&286206848099549185>
tranquil pine
i have the slope and a point for this one, but for some of the other ones i just...
show
full marsh
pallid birch
Is it 10th?
full marsh
10-13, but 10 first
pallid birch
You know the equation of a line?
full marsh
y=mx+b
pallid birch
Just put y, x(given coordinates) and m there to get b. Then write the whole equation.
full marsh
ok but what about standard form
pallid birch
You got b right?
Now the standard form is
y=mx+b
Just put m and b there not the co-ordinates
full marsh
that's slope-intercept form
standard form is ax + by = c
i need to know how to convert between them
pallid birch
Oh yes,
You're right.
standard form yields
by=c-ax
y=c/b-(a/b)x
Compare it to
y=mx+c
m will be -(a/b) and y-intercept will be c/b
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regal relic
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regal relic
can someone make sure i did it correctly
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@regal relic Has your question been resolved?
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regal relic
<@&286206848099549185>
pallid birch
Hi
regal relic
Hi
do you mind checking if my working is right? i'm self studying this chapter and don't have answers
pallid birch
Will do.
regal relic
thanks so much!!
final saddleBOT
@regal relic Has your question been resolved?
pallid birch
It seems correct
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@regal relic Has your question been resolved?
eternal zinc
Is there anyway to simplify this further?
final saddleBOT
Is there anyway to simplify this further?
eternal zinc
random forge
,r
eternal zinc
,rotate
soft zealotBOT
eternal zinc
The + 1 is not part of the squareroot
eternal zinc
soft zealotBOT
random forge
throw that into Wolfram and see if it's the same as what you have
,w simplify sqrt(1+x^2+sqrt(1+x^2+x^4))
soft zealotBOT
eternal zinc
Pretty diff ngl
random forge
yeah but they look pretty similarly simplified
safe flare
You can remove the radical from either the nume or deno by doing the conjugate, just only choose either one.
eternal zinc
You can remove the radical from either the nume or deno by doing the conjugate, ...
Mb 🙏
safe flare
👍
eternal zinc
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eternal zinc
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eternal zinc
Mb the answer I did was wrong. This is the final answer 😂
I forgot the squareroot
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reef mantle
Hello can someone help me with natural deduction ?
According to the teacher, we have to reason by absurdity, use efsq and finally the introduction to disjunction
final saddleBOT
Hello can someone help me with natural deduction ?
According to the teacher, we ...
@reef mantle Has your question been resolved?
ember fern
final saddleBOT
stray mesa
your forgot your chain rule
ember fern
your forgot your chain rule
how does that work again
stray mesa
let u = sinx
then y = 1/u, u = sinx
differentiate y in terms of u, differentiate u in terms of x
raven marsh
$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$
raven marsh
this is chain rule
stray mesa
you get dy/du = -u^-2, du/dx = cosx
raven marsh
see how it helps (how to define f and g)
stray mesa
**Death.cv**
i really have to learn how to do that
raven marsh
i really have to learn how to do that
latex isnt that hard fr
proven solstice
i really have to learn how to do that
highkey just use chatgpt to format for a bit then you'll pick it up naturally
ember fern
**Death.cv**
ohh is it chain bcz there’s a power to negative 1
raven marsh
what
stray mesa
ohh is it chain bcz there’s a power to negative 1
uhhh
i suppose that is one way to put it
raven marsh
you can let $f(x)$ be $\frac{1}{x}$
raven marsh
and $g(x) = \sin x$
raven marsh
**Death.cv**
now use this to figure out derivative
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jagged flare
find all integers (x,y) such that $xy+\frac{x^3+y^3}{3}=2007$
final saddleBOT
find all integers (x,y) such that $xy+\frac{x^3+y^3}{3}=2007$
soft zealotBOT
Skill_Issue
jagged flare
basic rearrsnging gives $x^3+y^3+3xy=6021$
soft zealotBOT
Skill_Issue
jagged flare
then... no clue
ocean surge
6021
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lament vale
maybe
Yes, that's a good idea
Now you can just look at prime factorization of 6020
Also note that one of the brackets is non-negative
So x+y-1 must be positive
jagged flare
do i check all factors?
lament vale
If there're not too many factors then yes, why not
jagged flare
theres 16
lament vale
x+y-1 must divide 2*6020
Actually 6020
jagged flare
x+y-1 must divide 2*6020
Actually 6020
right so its 24
lament vale
Well, that's feasible
jagged flare
jeez
lament vale
Perhaps a more elegant solution exists
jagged flare
i hope so
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sullen salmon
so here -a^2 is actually -a * a ? is that why they could took " a " common here?
formal trail
yes
sullen salmon
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sharp snow
?
sharp snow
Is there a valid solve for x
final tangle
no
twin beacon
if you try to solve for it, you’ll get x=4, and x cannot equal 4 because then we’d have 4/0 which is undefined so there is no solution
sharp snow
what if we Multiply the both sides with x-4 so the x-4 can rid of it so x+1=4
x=3
??
😅
twin beacon
you’d have x + (x-4) = 4 because of the distributive property
shut gazelle
what if we Multiply the both sides with x-4 so the x-4 can rid of it so x+1=4
x...
Plug in 3, and see if the left side equals the right side, to check
sharp snow
Plug in 3, and see if the left side equals the right side, to check
yes no answer
thanks to everybody
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willow ember
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ember trout
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hexed lance
hi can i have help solving this #3
final saddleBOT
hi can i have help solving this #3
tulip coyote
,rccw
soft zealotBOT
tulip coyote
Do you have any idea of how you'd make use of it? 
hexed lance
doesnt that just = 0 so its indeterminate?
tulip coyote
What's the "that" you're referring to?
hexed lance
ohh
tulip coyote
Anyways, any ideas?
hexed lance
take derivative?
tulip coyote
Well, not yet(!) we need it in a form we can use lopital's on first, and we aren't yet there (we don't have a 0/0 or ♾️/♾️ case yet)
tulip coyote
Yep, both would be nice 
hexed lance
how though
tulip coyote
Remembering that $a^b$ is (where it makes sense) the same thing as $e^{b\ln(a)}$ gets us most of the way there, if you choose $a$ and $b$ appropriately
soft zealotBOT
@tulip coyote
hexed lance
so e ^ 4sin (x) ln(x)
tulip coyote
Yep (but don't forget the 4 of course 
)
Anyways, this is closer to what we want, and we just need to check what $\lim_{x\downarrow 0} 4\sin(x)\ln(x)$ is
soft zealotBOT
@tulip coyote
tulip coyote
But is that yet in a "lopiatalable" form yet? 
tulip coyote
Anyways, this is closer to what we want, and we just need to check what $\lim_{x...
(whatever we find from here, we then just "e^" that of course)
hexed lance
isnt ln(0) undf
hexed lance
But is that yet in a "lopiatalable" form yet? <:pikathink:972100841748185208>
no
i think
soft zealotBOT
tulip coyote
Notice what happens as you decrease to 0
hexed lance
it comes like down
tulip coyote
Yep, basically we have $\lim_{x\downarrow 0} \ln(x) = -\infty$
soft zealotBOT
@tulip coyote
tulip coyote
So we have a limit of the form -♾️ * 0, which is still indeterminate, but not yet "lopitalable"
Do you know what we could do to make sure it can be "lopitalable"?
tulip coyote
Yep, one of those forms 
Why, it's a ping to the <@&268886789983436800>
Anyways, any idea of how to get it into the form we need @hexed lance?
barren hound
1+1 = 2 day mute, don't troll here
upper slate
<@&268886789983436800> why chartbit
tulip coyote
awww
Well, have you heard (or are aware) of "larger denominator, smaller fraction"?
(in other words, the larger (in absolute value) that u is, then the smaller (in absolute value) 1/u is)
tulip coyote
well, we have something that gets quite large in absolute value, and I hope you're happy that 1/(1/a) is just a, right?
hexed lance
yes
tulip coyote
Well then, does that give you any ideas of how you could get one of the forms we want? like 0/0 form maybe? 
hexed lance
wouldn’t it give us inf/ inf?
tulip coyote
Well, depending on how you want to use it, it could(!)
You can get either one, depending on what you'd prefer to work with
hexed lance
wait im kinda confused 😭
hexed lance
<:catThink:519997422194393108> well, we have something that gets quite large in ...
is it like a rule or an equation
tulip coyote
tl;dr "a rule"
(wait, are you asking about the "larger denominator..." thing, or the 1/(1/a) = a thing (or both)?)
tulip coyote
Well a short way to see why is that if you took $\frac1{1/a}$ and multiplied it by $\frac1a$, you'd get
[
\frac{1/a}{1/a} = 1
]
and so you know that $\frac1a \cdot \frac1{1/a} = 1$, so if you multiplied both sides by $a$, you get
[
\frac{\cancel{a}}{\cancel{a}} \cdot \frac1{1/a} = a
]
(editing)
hexed lance
would we get like 4 ln(x) over 1/x to get - inf over inf
soft zealotBOT
@tulip coyote
tulip coyote
would we get like 4 ln(x) over 1/x to get - inf over inf
And almost but not quite, we didn't have an x by itself (you could instead do 4ln(x)/(1/sin(x)) for inf/inf if you wanted)
tulip coyote
Well then, does that give you any ideas of how you could get one of the forms we...
(You could have instead done 4sin(x) / (1 / ln(x) ) for 0/0 if you wanted too)
Depending on what you would prefer to work with of course
hexed lance
ohh okayy
hexed lance
And almost but not quite, we didn't have an x by itself (you could instead do 4l...
wouldn’t i still get 0
tulip coyote
Get zero for?
hexed lance
derivative of 4ln(x) over derivative of 1 over sin x
soft zealotBOT
hexed lance
okayy
tyy
i have one last question and its on critical values 😭
i think im halfway through but im kinda lost
tulip coyote
Sure, which one
hexed lance
tulip coyote
Cool, the critical values you want between 0 and 9 inclusive, right?
hexed lance
yes
tulip coyote
And assumedly you're checking the denominator being zero for the points where it's undefined, and that's where you're stuck?
hexed lance
yess
tulip coyote
Well, you could e.g. try the quadratic formula/discriminant (for the discriminant, you should find it to be negative, so you know the denom is never zero)
hexed lance
why
tulip coyote
For the quadratic formula $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, working with $x^2 - x + 9$, you have $a = 1, b = -1$ and $c = 9$
soft zealotBOT
@tulip coyote
tulip coyote
Well, as I was about to get to-
You actually have the discriminant, the $b^2 - 4ac$ part working out as $(-1)^2 - 4(1)(9) = -35$, and so you have no real zeros
soft zealotBOT
@tulip coyote
tulip coyote
Cause you're then working out $\frac{1 \pm \sqrt{-35}}2$
soft zealotBOT
@tulip coyote
tulip coyote
Basically yep 
hexed lance
so my only critical value us 3?
tulip coyote
Yep
tulip coyote
If they're just asking for the critical values, you're basically done
If e.g. they ask for the coordinates or something, then you'd put it in the original
hexed lance
thank youuu
im all done
i hope you have the most wonderful day ur actually amazing 😭😭
tulip coyote
Awwww, hope you have an amazing rest of your day too 
opaque ivy
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opaque ivy
How did they arrive to cos?
placid halo
cos²x+sin²x=1
opaque ivy
Then how did they arrive to tan on this one? Pythagorean identity doesn’t seem to apply because it’s -1
soft zealotBOT
ColdTe²
placid halo
cos²x+sin²x=1
you still use this formula but divide by cos²x
opaque ivy
But how can you do that if there is only an addition operation?
obsidian nest
why |x| will never get to -1? are there no solutions in the complex world?
distant sleet
why |x| will never get to -1? are there no solutions in the complex world?
Just a product of the absolute value function I think
Complex numbers themselves have an absolute value which is their length
How would there be complex numbers solutions to the modulus function is the main question
The purpose of the function is to take an input and return a positive output so there can't exist a case where that becomes negative
It goes against the purpose of the function
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manic leaf
I've been stuck on this problem for a little while
show that there is no n >= 2 such that 2^n - 1 is divisble by n
final saddleBOT
I've been stuck on this problem for a little while
show that there is no n >= 2...
short talon
first show that it is true when n is prime
short talon
or 2^n == 2 mod n if n is prime
right, so 2^p - 1 = 1 mod p for a prime p, so it's not divisible when n is prime
now show that this implies the statement for all n
manic leaf
now show that this implies the statement for all n
alr this is kinda eluding me rn
2^p - 1 = 1 mod p, so theres some way i can represent n in terms of p?
short talon
2^p - 1 = 1 mod p, so theres some way i can represent n in terms of p?
yes, take the prime factorization, also hint: ||show that (2^a-1) | (2^ab - 1)||
warm jasper
yes, take the prime factorization, also hint: ||show that (2^a-1) | (2^ab - 1)||
what is a and b
short talon
what is a and b
any two numbers
warm jasper
that is also obvoius
note that we are done immeidately if n is prime
we are interested in nonprime n
manic leaf
wait hmm
wait im a little confused what if we just show 2^p1p2p3...pn
if every 2^pi is congruent to 2 mod n, thne 2^product of primes is congruent to 2 mod n?
warm jasper
no u need every 2^(p_i^e_i)
manic leaf
oh
short talon
actually, i was thinking that you don't need the entire prime factorization, you only need the smallest prime divisor
manic leaf
no u need every 2^(p_i^e_i)
but doesnt that not matter because if we any prime even if its repeated its still 2 mod p
short talon
but doesnt that not matter because if we any prime even if its repeated its stil...
this doesn't necessarily imply anything mod n though
short talon
so ok, take the smallest prime divisor p, you've shown that it divides 2^(p-1)-1 right?
assuming n divides 2^n - 1, it also divides 2^n - 1 as well
so show that it divides 2^gcd(n, p-1) - 1
manic leaf
this doesn't necessarily imply anything mod n though
im still a little confused about this though
lets say we have p1, p2
2^p1 is congruent to 2 mod p1
2^p2 is congruent to 2 mod p2
2^p12^p2 is congruent to 2 mod p1p2?
or maybe im misunderstanding a property of mod
short talon
oh this is probably true by crt
manic leaf
yes
short talon
ok, yeah i think that works
short talon
sure, it is a fact that if gcd(2^a-1, 2^b-1) = 2^gcd(a,b)-1, which you can see by taking the euclidean algorithm
then, by flt p | 2^(p-1) -1
and by assumption p | 2^n -1
so p | 2^gcd(p-1, n)-1
but gcd(p-1,n) = 1
since p is the smallest prime to divide n
short talon
lets say we have p1, p2
2^p1 is congruent to 2 mod p1
2^p2 is congruent to 2 m...
actually wait, then wouldn't 2^p1 * 2^p2 = 4 mod p1p2?
short talon
actually wait, then wouldn't 2^p1 * 2^p2 = 4 mod p1p2?
without crt, you can see this by taking phi(p1 * p2) = (p-1)(p-2), so then by euler's theorem 2^p1 * 2^p2 = 2^2 = 4 mod p1 p2
??
i got helpful?
after all these years?
short talon
think you could make yours works somehow, by taking gcd(n, phi(n)) but im not sure
anyway, does my solution make sense?
blissful torrent
final saddleBOT
blissful torrent
im pretty sure this is the answer, but i need to simplify 9 root 3
can anyone explain how to do that?? im not really sure
i know that root 3 is 9 so it can be simplified but i cant get my head around it
distant sleet
you made a small mistake
you went from sqrt(4) * sqrt(3) to 4 * sqrt(3)
and you can't simplify it any further
blissful torrent
ohh
i was following the tutorial video too closely i think
how would i write sqrt4 and sqrt3 together so i can add the terms together to get the answer
distant sleet
how would i write sqrt4 and sqrt3 together so i can add the terms together to ge...
it just becomes 2*sqrt(3)
then you have 5sqrt(3)+2sqrt(3)
blissful torrent
do you get 2 because the lowest factor of 12 is 2^2
or
im slightly puzzled on where 2 would be from if not that
distant sleet
im slightly puzzled on where 2 would be from if not that
Sqrt12 = sqrt(4)*sqrt(3)
What's the square root of 4
rocky tusk
🤔🤔
rocky tusk
What's the square root of 4
tough one
rocky tusk
definitely
is it solvable without a calculator?
…
@blissful torrent
$\sqrt(4) = $
$\sqrt{4} 😒
bruh
😒
why is that = $
yea i wanted the equals though
blissful torrent
hey sorry i was afk
blissful torrent
What's the square root of 4
ohh right 2
im just starting to get the hang of roots and surds so ya
so the simplified answer is 2 square root of 3 + 5 * square root of 3 which is 7 square root of 3
yep it was
thank you
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wind glen
yea i wanted the equals though
[ \sqrt{4} = ]
soft zealotBOT
w
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paper kite
Is it possible to draw a tangent and secant line from one point such that they are perpendicular to one another?
final saddleBOT
Is it possible to draw a tangent and secant line from one point such that they a...
cunning stirrup
well when the point is that far, not really
wind glen
idk if i read that right but
cunning stirrup
cunning stirrup
the two lines? of course
paper kite
i'm kind of trying to translate the problem into english
wind glen
they share a point
i suppose the intersection point could be the point of tangency? not sure what you mean
wind glen
wind glen
that doesnt look perpendicular but ig
cunning stirrup
sure. you could just consider them to be rays and have something like this
final saddleBOT
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paper bramble
A new car costs $24000. It loses 18% of its value each year after it is purchased. Determine the value after 30 months
final saddleBOT
A new car costs $24000. It loses 18% of its value each year after it is purchase...
paper bramble
i just wrote "let x be time in months" and "y-cost" but got lost from there
warm ether
no clue?
paper bramble
for the equation, i wrote f(x)= 24000(0.18)^(x/36) but i dont think thats right
warm ether
the 12 i agree fixes that part
but what does multiplying something by 0.18 do
percentage wise
paper bramble
oh it multiplies the percentage, i thought about subtracting it but im not sure if thsts right either
warm ether
elaborate
paper bramble
since the question says "it loses 18 percent" i thought that it would be 24000 - 0.18^(x/12) but i changed it to 24000(0.18)^(x/12) because for the examples my teacher provided she wrote equations like that
warm ether
that wouldnt make too much sense, but youre on kind of the right path at least
if i multiply something by 0.18 i am left with 18% of the original
i havent lost 18%, rather ive lost 82%
warm ether
100-18=82
paper bramble
oh ok thx
warm ether
if i want to model a loss of x% then we multiply by 1-x/100
eg here we would be multiplying by 1-0.18=0.82
paper bramble
oh ok so the equation would look like y= 24000(1-0.18)^(x/12)?
warm ether
yeah that should work, for x in months
tranquil flume
What is x?
paper bramble
oh ok then i sub in x for 30 to find the answer?
paper bramble
What is x?
i think the time in months
tranquil flume
I'm lil confused
paper bramble
heres my question "A new car costs $24000. It loses 18% of its value each year after it is purchased. Determine the value after 30 months"
AZO helped me figure out the equation
thank you azo btw
tranquil flume
30/12 is around 0.6 right?
paper bramble
my calculator says 2.5
paper bramble
yea i think
tranquil flume
I always hated how they taught percentage at school
I didn't understand why they multiply stuff
paper bramble
yea these concepts are pretty tough
im doing my best in adv function but its tough
thank you azo and diesel for helping me
warm ether
a percentage is a ratio out of 100
so 18% is 18/100 18 per 100, percent
so if i want 18% of a number, i want 18/100 of it
tranquil flume
Yeah and anything I see "of" that means multiply for me lol
final saddleBOT
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nocturne marsh
If given the quadratic equation $x^2-1=0$ which has roots $\alpha$ and $\beta$, then
$\alpha+\beta=0$ and $\alpha\beta=-1$
final saddleBOT
If given the quadratic equation $x^2-1=0$ which has roots $\alpha$ and $\beta$, ...
soft zealotBOT
solaris
drowsy crown
True
subtle urchin
I think it’s true
drowsy crown
Where is the question
drowsy crown
Don’t wanna get that late text so you gotta use latex
nocturne marsh
If I wanted to form a new quadratic equation with roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$, we can let $x=\frac{1}{\alpha}$. Then, $\alpha=\frac{1}{x}$. We can now substitute this value of $\alpha$ back into the original equation to get the new equation. Why does this method work?
soft zealotBOT
solaris
final saddleBOT
@nocturne marsh Has your question been resolved?
hushed saffron
**solaris**
You see the roots are
1/alpha and 1/beta
What is the sum of the roots
nocturne marsh
I don't wanna do it that way
There's an alternative way where you use a substitution into the original equation
I wanna understand why that actually works
final saddleBOT
@nocturne marsh Has your question been resolved?
pallid birch
You want to form an equation that has roots
1/alpha and 1/beta from the given equation whose roots are alpha and beta?
If I understood clearly
nocturne marsh
yes
I want to use the substitution method, not summing and multiplying $\frac{1}{\alpha},\frac{1}{\beta}$ and then making the substitutions for $\alpha+\beta$ and $\alpha\beta$
soft zealotBOT
solaris
pallid birch
Yes I can explain.
nocturne marsh
Go right ahead
pallid birch
Let's say
You have the equation
x^2-4=0.
You solve the equation and find the roots to be +2 and -2.
Now,
You wish to create an equation with the roots +1/2 and -1/2.
You know the general equation ax^2+bx+c=0
Substitute 1/2 and -1/2 there, find relation between a,b and c. You'll get c=-a/4 and b =0.
So the equation becomes
ax^2 -(1/4)a=0
x^2-(1/4)=0.
nocturne marsh
Okay
I suppose that makes sense, yeah
But in context of $\alpha$ and $\beta$ though, the reason we can substitute $\alpha=\frac{1}{x}$ is because $\alpha$ is a known root for which the equation = 0?
soft zealotBOT
solaris
pallid birch
Yes
nocturne marsh
How do we start with $x=\frac{1}{\alpha}$ though? Like what does that mean in the context of the equation? Couldn't $x$ mean any x value? Why specifically the root?
soft zealotBOT
solaris
pallid birch
The equation will only be zero when x is the root of the equation, not for any x values.
You want the root to be 1/alpha right?
nocturne marsh
Right, so essentially we are saying "let the new root be $x=\frac{1}{\alpha}$"
soft zealotBOT
solaris
pallid birch
Correct.
nocturne marsh
Then we solve for $\alpha$ to substitute back into the original equation because $\alpha$ is a root and hence the value of $\alpha$ would make the equation = 0.
soft zealotBOT
solaris
nocturne marsh
It's so interesting how that transforms the equation though
I can't tell if it's because I'm low on sleep or just overthinking it but this concept feels overly simple to be puzzled by
pallid birch
Yes you can create a new equations from given equation, having roots like 3/5 times or whatever times the root of the given equation.
If you know the roots of given equation.
nocturne marsh
That's so cool tbh
Thanks sm for the explanation 🙏
I can't believe it went over my head like that
pallid birch
Np
nocturne marsh
.close
final saddleBOT
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river ivy
HOw do I solve this question. im pretty sure the answer is D although I am not sure
final saddleBOT
HOw do I solve this question. im pretty sure the answer is D although I am not s...
zealous storm
HOw do I solve this question. im pretty sure the answer is D although I am not s...
is what u did not correct?
river ivy
is what u did not correct?
my friend got e
zealous storm
my friend got e
Well u got 18, which is lower than 21
and it works too
so if anything
it must be 18 or lower
instead of higher
675, 720, 768 is a valid geometric sequence
river ivy
ok tahnks i just needed verification on my work since not sure if my work was correct lol self doubt
.close
final saddleBOT
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trim lance
i need help idk where to begin with the rows
final saddleBOT
i need help idk where to begin with the rows
random forge
are you supposed to rref it?
severe canyon
I firstly suggest you to swap the first row with the third one, so that you have a leading 1 in the first row. This should make the calculations somehow easier
trim lance
I firstly suggest you to swap the first row with the third one, so that you have...
true
severe canyon
Then, after that swap, you can do R3 → R3 - 2R1 and R4 → R4 - 3R1
trim lance
Then, after that swap, you can do R3 → R3 - 2R1 and R4 → R4 - 3R1
then i can do 2R4-R1
?
or no wait R2/-2
and the rest i can sub in right
severe canyon
Now swap second row with fourth row
So that you have a leading 1 in the second row
Again, it's not necessary but I think it's quite useful if we do it
Try always to make calculations as simple as possible
severe canyon
Now you can do R3→R3 - 3R2 and R4→R4 + 2R2
final saddleBOT
severe canyon
so the gen solution is 1 2 -1 -2
Yep perfect
final saddleBOT
✅
trim lance
@severe canyon
This is for Y
and the dimension of the space is 2
do i need to say something else
severe canyon
do i need to say something else
I've never done geometry in R⁴ so I don't actually know how to show that those are planes
Let's wait for some other helpers
trim lance
Alrightt
final saddleBOT
@trim lance Has your question been resolved?
trim lance
<@&286206848099549185>
hoary dock
and the dimension of the space is 2
Saying this is enough, I believe
trim lance
Saying this is enough, I believe
because if i just say that it makes the problem short
which is weird
variables minus number of equations
hoary dock
Ah, you’re going to have to prove that the dimension of the solution space is 2
trim lance
how
hoary dock
variables minus number of equations
Which isn’t necessarily this
trim lance
oh with the equations that i did
hoary dock
Is this a linear algebra thing?
hoary dock
The quickest way is to construct a basis for X and Y
trim lance
how
hoary dock
The quickest way is to construct a basis for X and Y
Sorry this is a bit inaccurate, give me a se
sec
trim lance
kk
hoary dock
Suppose the rhs for these equations are 0
We’ll find that the solution set is a vector space of dimension 2
hoary dock
Right hand side
trim lance
oh kk
hoary dock
For any solution (x1, x2, x3, x4) to the original system of equations and a solution (y1, y2, y3, y4) to the system of equations with 0 on the rhs, then (x1+y1, x2+y2, x3+y3, x4+y4) will be a solution to the original system of equations as well
trim lance
hmm
hoary dock
It shouldn’t be too hard to verify, just add them together
trim lance
ok but that is if we suppose that the rhs is 0
hoary dock
Yes
hoary dock
The key here is to bring a solution that does actually satisfy the original case
Say for instance (7,0,3,0)
If (y1, y2, y3, y4) satisfies the equations with rhs 0, then (y1+7, y2+0, y3+3, y4+0) will satisfy the original case
hoary dock
The solution set for the original problem would then be of the form (7,0,3,0) + V, where V is the solution set to the rhs 0 case
Which is just V but shifted
We can verify V to be a plane
trim lance
so wait
trim lance
The solution set for the original problem would then be of the form (7,0,3,0) + ...
can for Y it would be (2,0,1,1) + V for example
hoary dock
can for Y it would be (2,0,1,1) + V for example
I think (2,0,1,1) won’t work here, but that’s the gist
hoary dock
maybe (2,1,1,0)?
final saddleBOT
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gray acorn
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gray acorn
how can an electron have a n of 4 1/3
final saddleBOT
@gray acorn Has your question been resolved?
white sphinx
how can an electron have a n of 4 1/3
it cant
gray acorn
it cant
oh ok
by the way by the depiction
does that mean
n=4 has a wave thing of like 4 per level
and n=5 is 5
or is the depiction not accurate
white sphinx
wdym
gray acorn
wdym
like n=5
here
has 5 bumps
wait im stupid for asking this even
NVM LOL tytyu
.close
marble flax
Paper cost is less than binding cost by what % ?
final saddleBOT
`Paper cost is less than binding cost by what % ?`
marble flax
How to solve these types of questions?
vernal hill
i believe u just do 30-15 to find the percent
marble flax
thats wrong answer
marble flax
yes
raven marsh
`Paper cost is less than binding cost by what % ?`
Alright first you know that cost of binding is 2x the cost of paper
raven marsh
You got the base wrong
Its (30-15)/15x100
As we are talking about the percentage of the paper cost
marble flax
isnt it similar to this question
marble flax
Yes but the phrasing means base is paper cost
how come 😭
raven marsh
Idk how to explain 😭
marble flax
final saddleBOT
@marble flax Has your question been resolved?
final saddleBOT
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random apex
Highschool math problem. Just started studying roots in limits. I've looked into solving this with a combination of a^2 - b^2 = (a-b)(a+b) and a^3 - b^3 = (a-b)(a^2 + ab + b^2), but the powers get very big and uncomfortable to work with. Is there a quicker method/trick to get out of the this [infinity - infinity]?
final saddleBOT
Highschool math problem. Just started studying roots in limits. I've looked into...
opal osprey
sure! But the answer is kindof tricky!
random apex
sure! But the answer is kindof tricky!
As long as it doesn't involve something like l'hopitals (we haven't got there), I'm willing to hear another idea
opal osprey
take the cubeth term common in the first term and the squareth term in the second
opal osprey
As long as it doesn't involve something like l'hopitals (we haven't got there), ...
of course…, mahbe a little expansions though…
random apex
of course…, mahbe a little expansions though…
We'll see, i'll follow your idea
opal osprey
mhm… now take the n common from both the terms… leaving us with two (binomal raised to fractional power) terms
random apex
opal osprey
my idea was to use the binomial expansions for them..
random apex
so I end up with something like this?
opal osprey
Yes… i was thinking maybe take the cube and square out of their respective roots
random apex
English is not my first language so I hope I don't misunderstand math terms.
alright
opal osprey
English is not my first language so I hope I don't misunderstand math terms.
Oh, its fine… I can help out. Dont worry
opal osprey
yes
so you know how to use a binomial expansion or do you need a proof of it using taylor’s expansions?
if you want… ill give you the proof of taylor’s too!
fringe sigil
if you want… ill give you the proof of taylor’s too!
(unrelated to him, can u give it to me after u clear his doubt)
opal osprey
(unrelated to him, can u give it to me after u clear his doubt)
ah of course! Add me!
random apex
Sure, I know about binomial expansions (something with Pascal's Triangle numbers as coefficients), but I have no idea who taylor is (taylor series related?, I haven't studied those)
random apex
Sure, send it, I'd like to see it. If I can't find another way to solve it that is more at my class' level then I'll just have to go with the original idea.
opal osprey
alright… give me five minutes, k?
random apex
alright… give me five minutes, k?
Sure.
opal osprey
here
now assume f(x) is (1+x)^n
youll get its binomial expansion using this!
do you get it?
random apex
I see, but for a class that is yet to study derivatives, it's not something I can present as an alternative solution. Thanks for the help, but it seems like going with something like a^6 - b^6 expanded as (a-b)( a^5 ... ) is the only thing I can go for at my level.
opal osprey
I see, but for a class that is yet to study derivatives, it's not something I ca...
oh alright… here, we learn taylor’s expansions in limits itself… sorry, i dont have any other solution!
my teacher gave us the exact problem you showed me two months ago!
random apex
It's alright, I was really caught onto the fact that n^3 + 3n^2 is the first half of (n+1)^3. I was wondering if there was a trick somewhere to avoid the headache.
random apex
my teacher gave us the exact problem you showed me two months ago!
What an awesome coincidence.
opal osprey
yeah!
random apex
Alright, well, if there is nothing else for me to do I guess I'll get to writing, thanks for the help though.
.close
final saddleBOT
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obtuse schooner
hiya, could someone help me out with what the letter o means in this fomrula? My best guess owuld be that its an integral but can't find anything on google about it
final saddleBOT
hiya, could someone help me out with what the letter o means in this fomrula? My...
@obtuse schooner Has your question been resolved?
obtuse schooner
awesome, thanks
is there any literature on it? I believe you but its just weird to me so i'm curious as to why its the o
candid hull
is there any literature on it? I believe you but its just weird to me so i'm cur...
someone's software screwed up the rendering
it's likely a displaying bug, not something intended
final saddleBOT
obtuse schooner
average uni experience hahaha, thanks though!
candid hull
you could try looking for other ò in your uni documents I guess
you'll have more clues then
final saddleBOT
Available help channel!
terse crypt
do u guys know a book that covers:
- set
- logic
- real number and polynomial
- function
- transcendental functions i.e. log and expo
- trig
- complex no.
- matrix
- sequence & series
- calculus
- analytical geometry and vector
- statistics, distributions, combinatorics, and probability
- number theory
- Linear Programming
final saddleBOT
do u guys know a book that covers:
1) set
2) logic
3) real number and polynomial...
fathom kayak
thats a lot of topics
fathom kayak
yeah
short term u are better off spamming practice questions
do you have example exam papers to study from?
fathom kayak
do the solutions come with it?
fathom kayak
well the one thing this discord is good at is explaining
you have a lot to work with: questions, solutions if u dont understand, this discord and ur own will power
do the questions, if u dont understand try look at solutions and understand the steps then do it again
watch youtube videos for further guidance or ask in here if u dont understand the solutions
there are a few books i could find for u but i would honestly recommend doing the questions
especially since ur on a time crunch
robust minnow
i have exam on sunday
Make sure you grind and don’t have any distractions at all, it’ll be possible to completely cover all of those (I know because I did the exact same 💀)
Also check out Organic Chemistry Tutor on YT (he explains math topics so you can understand the core concepts)
fathom kayak
yah
