#help-10
warm shaleBOT
dldh06
nocturne minnow
And that's not it
nocturne minnow
Yes
elfin tangle
how would I do domain?
nocturne minnow
Do you know what domain is?
cerulean tulip
yeah
obtuse pebbleBOT
@elfin tangle Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
carmine token
obtuse pebbleBOT
carmine token
would i need to multiply the numerator and denominator by 1/x^2?
fierce lagoon
No
carmine token
hmm
fierce lagoon
You just cancel low order terms
Like if you notice
$$\lim_{x\to\infty}\frac{2x^3}{3x^2-4}\Rightarrow\lim_{x\to\infty}\frac{2x^3}{3x^2}$$
warm shaleBOT
Umbraleviathan
carmine token
so we take out the -4
i'm just confused when we would multiply
since my professor's work multiplies on some problems @fierce lagoon
fierce lagoon
Not sure why
You can just cancel low order terms
Since they do jackshit against a literal infinity
carmine token
You can just cancel low order terms
what would be after taking out the -4?
fierce lagoon
Simplifying
carmine token
from that i evaluate what happens when i plug in infinity?
fierce lagoon
Yeah
carmine token
it keeps going up as i increment x with 1,2,3 etc
so that means my answer is infinity
fierce lagoon
Yeah
obtuse pebbleBOT
fierce lagoon
(Yeah)
obtuse pebbleBOT
Available help channel!
flint smelt
Where'd I go wrong with B?
obtuse pebbleBOT
Where'd I go wrong with B?
royal basin
g(0) should be 18
also keep in mind the diff between horizontal and vertical distance
flint smelt
I'm not following that -- like g(theta) should equal just the number 18?
harsh remnant
The question told you that Stephen started at 3 o'clock
flint smelt
I'm also not understanding that, this is my first few days of trig. The example for this we were given in class just had us add whatever the height from the ground was to the end, why/where is this different?
flint smelt
I think I'm missing some underlying thing, as I've also been having issues reading graphs like what you've drawn. I'm going to ask my professor, thanks all for the help though
.close
turbid sail
obtuse pebbleBOT
turbid sail
Could someone give me a hand with d
royal basin
@turbid sail try writing the equation as x^2 = z^2 - y^2
something may jump out at you
turbid sail
It’s a difference of squares?
turbid sail
something may jump out at you
Or is there more to it?
royal basin
there is a little more to it than just saying "it's a difference of squares"
turbid sail
it’s a difference of squares that’s result is odd soooo that’d mean it’d have an odd and even term ?
royal basin
i suggest that you actually write some things down.
turbid sail
i suggest that you actually write some things down.
Is that wrong ^ I’m not sure what else I can spot
royal basin
what you said is too slippery to be right.
royal basin
no, you've pulled some bullshit
you've set x = 2k+1 that's ok
but if you now set z to 2k+1 you're asserting z=x
i mean i guess like
with the problem statement as written that does actually work
take z = x and y = 0
since nobody said y and z had to be positive
royal basin
"Take z = x and y = 0; we have x^2 + 0^2 = x^2. QED."
turbid sail
Oh so I guess that would just work for all x
As there is nothing making y leave 0
because of the statement having there are integers y that satisfy x opposed to the other way
yeah?
sorry I just want some intuition to the problem
obtuse pebbleBOT
@turbid sail Has your question been resolved?
brave bramble
No that does work. Give me an odd integer x. I'll give you integers y and z such that x² + y² = z²
Of course, those integers can be y = 0, z = x.
Is there, uh, supposed to be more to it than that?
brave bramble
I think Ann was thinking about factoring x, and using that to fill in the difference of squares
Which, maybe is what the question intends? Idunno
turbid sail
I think Ann was thinking about factoring x, and using that to fill in the differ...
Factoring x?
brave bramble
Nvm, I actually don't know how it'd go! Ignore me.
brave bramble
Ah neat. Yeah, that works.
turbid sail
Idk how they come up with it tho
turbid sail
Oh yeah ?
brave bramble
Oh yeah!
turbid sail
what’s wrong with it
royal basin
Which, maybe is what the question intends? Idunno
yeah but malicious compliance tho
turbid sail
yeah but malicious compliance tho
I wasnt sure what you wanted me to do I’m sowy
royal basin
i can repeat myself
the statement you've been tasked with proving can be proved by presenting the example (y, z) = (0, x)
that's it
turbid sail
yeah but malicious compliance tho
Don’t be so hard on me I’m just trying to learn 😔
.close
slender kindle
obtuse pebbleBOT
slender kindle
well i was able to get r=ax-at(p)
p is perpendicular distance from the origin
hang on ill also send a lil working
another small error
dammit
wait
.close
vestal quarry
hello guys can any one explain why i can't use help-3,8,11,12,-----------25 ?
past fog
.close
feral island
Can someone explain what happened here?
obtuse pebbleBOT
Can someone explain what happened here?
fading musk
completing the square
like if I'm factoring x²+4x
you +4on both sides
you get (x+2)²
feral island
I understand that part
But I don’t get where did the 25/576 came from
Wait
Nvm I got it
thanks!
feral island
cuz have 2ab
Do you mind if I ask one more question?
fading musk
sure ok
feral island
or will a be 0
fading musk
b is -5/24
feral island
But isn’t x 1?
fading musk
a is x
feral island
Than do I just replace x with 1 in the formula
feral island
oooh
feral island
Do you just divide - 5/12 by 2?
fading musk
yes
obtuse pebbleBOT
@feral island Has your question been resolved?
harsh remnant
Did I do this correctly?
obtuse pebbleBOT
Did I do this correctly?
gilded needle
what's your rationale for using the derivative here? wouldn't it just be |x(5) - x(2)|?
harsh remnant
what's your rationale for using the derivative here? wouldn't it just be |x(5) -...
Because my problem sets are related to derivatives
gilded needle
x'(5) - x'(2) is the difference in velocity between time 2 and time 5, i'm not sure if you can relate that to distance traveled
harsh remnant
harsh remnant
x'(5) - x'(2) is the difference in velocity between time 2 and time 5, i'm not s...
So like this?
gilded needle
yes
gilded needle
presumably we assume that the position is continuous?
i.e the particle can't jump instantaneously?
if so then x(0) would have to equal the limit of x(t) as t->0, which is 0
harsh remnant
So we do the same as above?
gilded needle
yeah in general, assuming we're talking about NET distance traveled, it's just |x(t_end) - x(t_start)|
net distance means we don't count any backtracking
in your examples there's no backtracking anyway, as x'(t) is positive for the time range of interest
if you do want to count backtracking (i.e. if the particle moves from x=0 to x=2 and then back to x=0 you want to call that distance 4, not 0) then you would integrate |x'(t)| (the absolute value of the velocity)
obtuse pebbleBOT
@harsh remnant Has your question been resolved?
obtuse pebbleBOT
https://artofproblemsolving.com/wiki/index.php/Wilson%27s_Theorem
white urchin
can anyone explain me which part of FLT tells that? website links may be helpful
wooden cipher
this has to do with multiplicative order
white urchin
*sometimes helper get stuck too
wooden cipher
white urchin
this has to do with multiplicative order
like what?
white urchin
https://www.youtube.com/watch?v=KTnt0uW6qss
thanks
wooden cipher
yeah np
oh wait thats different
anyway do you know what multiplicative order is?
kind hawk
FLT says that x^(p-1)=1 mod p for all nonzero x
Subtract 1 from both sides and you get your polynomial
obtuse pebbleBOT
@white urchin Has your question been resolved?
white urchin
anyway do you know what multiplicative order is?
nope, is it like multiplicating consecutive number?
kind hawk
It's the smallest positive exponent k such that x^k=1
white urchin
It's the smallest positive exponent k such that x^k=1
oh...
white urchin
FLT says that x^(p-1)=1 mod p for all nonzero x
yes but it doesnt answer my question
wooden cipher
mulitplicative order is always a divisor of the totient of a number (usually its just the totient itself)
the totient for a prime is (p-1)
kind hawk
yes but it doesnt answer my question
Why not?
It says that x^(p-1)-1=0 for all nonzero x
Which is what you want
white urchin
mulitplicative order is always a divisor of the totient of a number (usually its...
OH
i get it
wooden cipher
im surprised lol
white urchin
i know some of those totient
wooden cipher
glad you know totient
white urchin
It says that x^(p-1)-1=0 for all nonzero x
yes, i just realized it now
thanks all @wooden cipher @kind hawk
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
timid silo
hi all, working this problem:
obtuse pebbleBOT
hi all, working this problem:
timid silo
not seeing what they did between the first two steps shown
clearly they are doing some factoring to the 11k+3 term
knotty crow
they did:
11k = 12k - k
3 = 4 - 1
timid silo
ok, is there any intuition on how you would see or know to do that? we're trying to show that this (the RHS) is equal to another expression and i suspect that is what's guiding that?
i suppose by getting that 12 you can then factor further the other terms
devout sable
Maybe looking at question will help get better content
timid silo
high lily
they completed the cube
timid silo
right then you can factor out 4 from each term
high lily
note that generally that might not actually help with factorisation, but here it does
high lily
yes
timid silo
why does it then become positive again by the end
oh they factor out a (k+1)
?
how can we see or know that first term in blue box will factor to second term
devout sable
how can we see or know that first term in blue box will factor to second term
More experience in maths, i.e more practice till you get comfortable with it
(I’m certainly not experienced enough tho)
daring bluff
Would anyone be willing to explain this to me? I can’t understand the problem one bit.
obtuse pebbleBOT
Would anyone be willing to explain this to me? I can’t understand the problem on...
daring bluff
And it says it’s a test, it’s not. It’s a summer school program and everything is just considered homework. Everything is called a test however
soft pelican
And it says it’s a test, it’s not. It’s a summer school program and everything i...
create 3 functions and combine them
proud delta
tranquil bronze
obtuse pebbleBOT
@daring bluff Has your question been resolved?
scenic sable
can someone explain to me why my teacher set 4/3 = 30/x like where did the 30 come from??
obtuse pebbleBOT
can someone explain to me why my teacher set 4/3 = 30/x like where did the 30 co...
high lily
30 is the perimeter of the original polygon
scenic sable
oh
but why would she set 4/3 = 30/x
if she wants to find the missing side length of PQRST
shouldnt she set 4/3 to 8/x
high lily
where does it say that they want the missing side lengths of PQRST
scenic sable
idk my teacher was solving for x so thats what i thought
scenic sable
no
high lily
ok, it may be a bit unclear but their x represents the perimeter of PQRST
scenic sable
ohh
high lily
they can use the scale factor to determine the perimeter directly (given the perimeter of the first shape) without having to calculate the lengths of individual sides
simple siren
I am asked to prove P(A U B) = P(A) + P(B|A') * P(A') . Can someone point me towards the theorem/property/rule that is the key to prove this?
obtuse pebbleBOT
I am asked to prove P(A U B) = P(A) + P(B|A') * P(A') . Can someone point me tow...
simple siren
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
still jackal
hi
still jackal
Can someone make me understand how to solve this graph
Because I don't understand it at all
Please
still jackal
But please how do you know that
I don't understand
I get that Q= IT
But I really feel dumb since I don't understand what's the relation between the area and
Like how does the area make sense?
prisma turret
Have you seen integrals before?
still jackal
Yeah a bit
I just want you to explain to me as if you will explain to a child
Because I don't wanna talk about integrals
Isn't there any way to explain it but that?
Are u in high school?
I am in high school
prisma turret
Well.......let's say this was a flat lined graph
And instead of I(t) it was V(t). v for velocity (or speed)
A straight line at y=2
This would mean you're going at a constant speed of 2 m/s for all time
But what if we wanted to know how much distance you're covering in that same time?
How would we find it?
If you're going 2 m/s, how far do you travel in say 6 seconds?
Well you refer to the distance equation d=rt
(which looks an awful lot like q=it)
It's a simple multiplication problem, but that multiplication is represented by the AREA under the flat line from 0 to 6
And that AREA is found by multplying (2 m/s)(6 s)
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
timid silo
Suppose $16,537.50 in interest was earned after $75,000 was invested at a simple interest rate of 3.15. How long (in years) was the money invested?
obtuse pebbleBOT
Suppose $16,537.50 in interest was earned after $75,000 was invested at a simple...
@timid silo Has your question been resolved?
tranquil bronze
timid silo
Thx
turbid sail
obtuse pebbleBOT
warm shaleBOT
turbid sail
Doesn’t the remainder technically need to reach 0 for the algorithm to be complete?
timid silo
by "complete", you mean terminate, yes
but we don't need to find the gcd, we only need to relate the gcd to gcd(a,b)
timid silo
Does this constitute as a solid proof for the question
so for the last line, you want to make some deduction on why this means that the two gcds are equal
timid silo
the euclidean algorithm says that the gcd is the same as the gcd between the quotient and remainder
the quotient is b, remainder is a
turbid sail
Oh well yes
Haha
I no smart
Yeah so that makes sense
So it’s essentially just proven from that
timid silo
yup :)
timid silo
I no smart
nah, its just confusing at the beginning, you will get used to it soon
obtuse pebbleBOT
timid silo
yw :)
obtuse pebbleBOT
Available help channel!
vital void
How do I make an equation that plots values in the scribbled bounds?
obtuse pebbleBOT
How do I make an equation that plots values in the scribbled bounds?
vital void
Is this a massive integral?
sour granite
How do I make an equation that plots values in the scribbled bounds?
The region outside the ellipse is 3x^2 + y^2 > 0.9, and on top of that you need to restrict x and y to be between -1 and 1
vital void
How would I deterministically do it
like if I were to choose a random x value, it would then choose another random y value that works
sour granite
Is this a massive integral?
You can pick any $(x,y)$ that satisfy all 3 of the following
$\begin{cases}
3x^2+y^2>0.9 \
-1<x<1 \
-1<y<1
\end{cases}$
warm shaleBOT
π=√g
sour granite
like if I were to choose a random x value, it would then choose another random y...
If you want you can solve for y in the first equation for that
frozen parrot
winged ibex
how does -3/8e^y come when doing x to x^2
obtuse pebbleBOT
how does -3/8e^y come when doing x to x^2
winged ibex
@slim cove ?
winged ibex
.close
prime scarab
Can somebody help me solve this matrix as am getting stuck with using the gauss Jordan elimination algorithm
obtuse pebbleBOT
Can somebody help me solve this matrix as am getting stuck with using the gauss ...
buoyant lark
slim cove
If somebody can still help me out
I can't quite read your last line, what does it say on the left?
slim cove
what do you mean by r2r1 - 5
prime scarab
And changed it to r2
slim cove
that's not a valid operation
the only three elementary row operations are
- switching the order of two rows,
- scaling a row, and
- row replacement
prime scarab
What I am trying to do their is combine with first row times second row and -5 to it
slim cove
you can't multiply one row by another row
that's like multiplying (2x + 4y)(3x + 8y) = 6x + 32y
It doesn't work
slim cove
you can't just subtract by 5
you need to subtract by 5 times row 1
that's like if you have the equation 3x + 4y + 3z = 1
and then you just subtract five from everything to get -2x - y - 2z = -4
it doesn't work
prime scarab
Did matrix in calculus 3 but not like this been a long time and watched YouTube videos
Yes
Next thing I am thinking is r2+3r1 to r2
Can get the middle to get to 1
Does not work as this part I am stuck
slim cove
hmm these numbers are rather annoying, but you could just scale r2 by -1/7
prime scarab
Keep on wondering if possible to solve these matrix to get a different way
prime scarab
Did and it worked
slim cove
cool
nocturne minnow
Are you trying to get it into RREF?
nocturne minnow
At first you add r3
nocturne minnow
So extremely stupid question, but are you doing this problem as practice for RREF?
prime scarab
Yes
nocturne minnow
Alright, I was just asking because RREF is just all 1s on the diagonal and since you're not solving a system, technically you could have just bypassed the work and wrote 1s down the diagonal
prime scarab
Sorry and using gauss Jordan elimination algorithm
Trying to solve for the identity
nocturne minnow
Overall, looks good
prime scarab
Watched YouTube videos on how to do this stuff and all
Feel like brain now goes dead
And am stuck with this all the time
Am using krista king to do differential equations to do this stuff and felt like cannot do this stuff
Thanks
@slim cove thanks
slim cove
no problem! you can do this!!
prime scarab
Had not done this type of math in like 7 years
nocturne minnow
It's like riding a bicycle. Once you get started again, it comes back
prime scarab
Okay as do get it now and wll
Thinking about doing linear algebra as want to study mechanical engineering
Thanks again as do want to get good at math
Think Eric is better at the differential equations than I can do it
slim cove
nah man with enough practice, you'll be better than I ever will
prime scarab
Okay and thanks for the encouragement
nocturne minnow
Thinking about doing linear algebra as want to study mechanical engineering
Just out of curiosity, you said you haven't done math for almost 7 years but want to do ME? Did you decide to not attend college for a few years?
prime scarab
Am doing cyber now as a job but have no interest in it as realized on how a stupid career field it is as do 4 years in college than 2 years of self training as college screw me over
Thank you
.close
timid silo
Just got this question wrong, answered False. could someone explain why this is true? I would assume that this would only be true if it was (abc)^2 instead of ((a^2)(b^2)(c^2))
obtuse pebbleBOT
Just got this question wrong, answered False. could someone explain why this is ...
timid silo
and that was my though proccess in choosing false
nocturne minnow
Because $(abc)^2 = (abc)(abc) = a^2b^2c^2$
warm shaleBOT
dldh06
deep brook
I need someone to teach me composite functions and help me with this problem
deep brook
This is the only problem that I dont understand how to do
timid silo
f(g(x))
try to replace x in f(x) by g(x)
first one is f(g(x))
f(x)
so its the first one
deep brook
I tried the first one first and found out its not that
timid silo
replace x in f(x) by x
deep brook
the question is a little tricky and says which would NOT be a possible function for f(x)
deep brook
yeah i realize now
deep brook
is there any other ways to work this problem out?
timid silo
its easy
deep brook
other than plugging it in sort of?
timid silo
always do f(g(x))
deep brook
I just thought I was doing it wrong lol
timid silo
wat
timid silo
wdym
deep brook
deep brook
ok my bad
timid silo
u good now?
timid silo
gl
wooden cipher
(8C4)(10C4)/(18C8)
Idk what that becomes
Sure
Oh this one is not fun
And i cant really think rn
Oh wait you choose 4
Not ss bad
So you have two cases and you add them
4cars: (6C4)
2trucks, 2 cars: (7C2)(6C2)
I believe thats 15+21×15=22×15=330
No prob have a good one :)
obtuse pebbleBOT
buoyant hull
Would I have to do $\frac{1}{R_1} \times (T_i - T_1) = q$ for each?
warm shaleBOT
rawlini
buoyant hull
Or i am missing something?
Because i dont understand how im suppose to make a matrix
<@&286206848099549185> (sorry)
dapper bloom
So T_1 thru T_3 and q are unknown but every other value is known right?
buoyant hull
Yea
dapper bloom
So sub out q in each one
buoyant hull
I dont follow
dapper bloom
To write each in the form (1/R)(T-T')-q=0
buoyant hull
Gotit
dapper bloom
Distribute out the 1/R
buoyant hull
Where does the = 0 come from?
dapper bloom
Subtracting the q
buoyant hull
oh duh
dapper bloom
Well I guess some of them you won't write =0
The ones with T_i and stuff for ex
But ignoring those
(1/R)T-(1/R)T'-q=0
Is how the ones not containing T_i or T_o would look
buoyant hull
Understood, writing them out rn
dapper bloom
Basically you just want to get the unknowns T_1,T_2,T_3,q isolated on one side is all.
You'll have four equations in four unknowns
buoyant hull
then just setup a matrix from the system?
dapper bloom
Yeah pretty much
buoyant hull
and substiute t_i and t_o in for the ones it contains?
dapper bloom
T_i and T_o have numerical values given
buoyant hull
Yeah thats what i meant
buoyant hull
So when writing the matrix out, if one equation doesn't contain a certain t value I just put a zero since it'll have no affect on the others
buoyant hull
ok
Also, you mentioned the ones containing t_i and t_o won't be equal to zero, why? and what would they be equal to
dapper bloom
I hope this is readable
I'm saying write your four equations in this form
With the unknown variables on the left
And any constants isolated on the right
So you will have to find the A_nm and a_k values
buoyant hull
i just got more confused
so i just wrote the equations out. let me post a photo
So I move the t_i to the right?
for equation (a) I can evaluate $$\frac{1}{.035} \times 23$$
warm shaleBOT
rawlini
dapper bloom
The corresponding matrix equation for this will have the form
dapper bloom
Yeah the parts involving t_i and t_o will evaluate to a constant that you will place on the rhs
It's pretty tedious to do all this so I'm just outlining how I'd approach it I guess. 
buoyant hull
im still really confused but i'll just go to office hours trmw b/c i barely understand half this shit
dapper bloom
Idk why you have ? here?
buoyant hull
b/c u said that the ones with t_i and t_o won't be equal to 0
dapper bloom
My bad
dapper bloom
I mean after u write it in this form
dapper bloom
Stuff like (1/R)(T_i-T_1)=q will give you (1/R)T_i-(1/R)T_1-q=0 ofc.
buoyant hull
yeah
dapper bloom
Which is fine, just means some of ur matrix entries will be zero too
dapper bloom
For checking ur work the values you get at the end for T_1,...,T_3 and q you can just plug back into ur starting equations btw.
So you don't have to feel too unsure of your work lol
buoyant hull
For checking ur work the values you get at the end for T_1,...,T_3 and q you can...
wait can you explain this?
dapper bloom
You'll know if you did it wrong because the values you get will be all messed up
dapper bloom
Yep exactly
buoyant hull
well let me write this in matlab and see how wrong i am lol
dapper bloom
You can check if T_1,T_2,T_3,q really solves your system at the end.
buoyant hull
so i got
22.6649
-16.5853
-20.3829
9.5732
they already look wrong
lemme see if they actually are
holy shit
i got 0
err actually 0.001 but im in engineering so its good enough
thanks man
i have a couple more problems similar to this one, do u have any tips on how to solve em?
Just like generic tips
.close
wanton dagger
How could I do this question ?
obtuse pebbleBOT
How could I do this question ?
wanton dagger
I was thinking of using the formula : a=net force/mass
But I guess that’s not going to work
Is some information missing or something?
copper latch
Yes ig frictional constant
copper latch
frictional force f= μmg
uncut robin
can't you find μ, you got f and mg
feral island
What did I do wrong for d?
obtuse pebbleBOT
What did I do wrong for ***d***?
royal basin
also the = signs on the start of each line should really be <=>
also what happened in line 4?
hybrid gull
Line 3 to Line 4 you gotta be careful when adding to complete the square
fading musk
why is equal sign on left side
royal basin
zeta function
thank you for confirming that you cannot be reasoned with
obtuse pebbleBOT
@feral island Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
rough wedge
hey
rough wedge
so, how do I know when to use ax^2 + bx + c
and when to use ax^2 + b
to find the pattern?
high lily
you should usually start with
ax^2 +bx +c
unless you have additional information
rough wedge
you should usually start with
ax^2 +bx +c
unless you have additional information
oh okay
i just don’t understand how I would have known that in example 1
it only says so in the answer
high lily
technically you shouldn't start with that
they just started with one of the simpler quadratic relations and had pre-knowledge
timid silo
Just wanted to ask, is any algebraic expression - like 2x+3(nothing specific here, just picked a random expression) - a function ?
obtuse pebbleBOT
Just wanted to ask, is any algebraic expression - like 2x+3(nothing specific her...
timid silo
Because from what I understand functions give output values for input values, so while there isn't an = sign in this expression, there can be one if we give a value to the x.
kind hawk
at first it's just an expression
we can define a function f(x)=2x+3 using this expression
but this function is something else
(and there are other technical details on what a function actually is, in particular stuff like what inputs and outputs we allow. also not every function has to "use" one of these algebraic expressions to calculate the output given an input)
obtuse pebbleBOT
@timid silo Has your question been resolved?
timid silo
(and there are other technical details on what a function actually is, in partic...
Alright.
gilded topaz
obtuse pebbleBOT
timid silo
suppose n has a square divisor, say $p^2\mid n$, then show that $p\mid \varphi(n)$ and gcd(p,n-1)=1
warm shaleBOT
Toby
obtuse pebbleBOT
@gilded topaz Has your question been resolved?
gilded topaz
I dont get it
obtuse pebbleBOT
@gilded topaz Has your question been resolved?
timid silo
I dont get it
suppose there exists a prime p such that $p^2\mid n$, in particular, $\nu_p(n)\geq 2$. Then $$p^{\nu_p(n)-1}(p-1)=\varphi(p^{\nu_p(n)})\mid \varphi(n)$$, in particular, $p\mid \varphi(n)$. But note that $n-1\equiv -1\mod p$, so $\text{gcd}(n-1,p)=1$, in particular, $p\nmid (n-1)$. Thus $\varphi(n)\nmid (n-1)$.
warm shaleBOT
Toby
gilded topaz
what does $v_p$ mean?
warm shaleBOT
FireBlazer
timid silo
largest power of p dividing n
gilded topaz
Okay, thank you!
timid silo
yw :)
elfin tangle
obtuse pebbleBOT
brisk tartan
elfin tangle
Juicy, this is someone else's help channel
Can you help with mine?
timid silo
Can you help with mine?
I'm trying to picture it, can't guarantee I'll be of any help until I figure it out for myself
timid silo
Other than this badly drawn diagram I can't really help
elfin tangle
Okay cool know the diagram - but how would I determine the function?
coarse ibex
If you make it start at (0,0) and end at (12,0) (since the length is 12m) then you know the function looks like $f(x)=ax(x-12)$ since 0 and 12 are roots.
warm shaleBOT
Morrow
elfin tangle
But what about the heights and stuff?
coarse ibex
That gives you a point on the parabola (well, 2, but it's really only information about one point).
So then you can solve for the appropriate a.
elfin tangle
Explain pls 😭 i dont get it
coarse ibex
This is what the function looks like.
https://www.desmos.com/calculator/23kgs1qojr
You need to figure out what the correct value of a is. You can figure that out from the information that at x=2 the height, f(x), is 8.
coarse ibex
I recommend making the start at 0 rather than centering it
Makes things a bit easier
elfin tangle
This is what the function looks like.
https://www.desmos.com/calculator/23kgs1qo...
The height is 8?
timid silo
I thought ax(x-12) would already be centered
Welp once again Morrow has me beaten to the punch, I'll go to another help channel
coarse ibex
The height is 8?
Right, the height at 2 should be 8. You can probably figure out what a should be by just playing with the slider, but you can solve it algebraically by plugging in x=2 and f(x)=8 and solving the resulting equation.
elfin tangle
Where would the 8 go in the equation?
coarse ibex
8 is the value of f(x)
Meaning they give you f(2) = 8, so expanding that out gives a * 2 * (2 - 12) = 8
elfin tangle
im so confused LOL
coarse ibex
Okay, so do you understand why the parabola is of the form f(x)=ax(x-12)?
elfin tangle
No 😭
coarse ibex
Alright
It's because we want a quadratic that zeroes out at 0 and at 12, since our bridge is 12 meters long. x(x-12) is a quadratic that does that. But then we can still have a constant factor there to scale the height of the bridge up and down, and it won't affect where our function zeroes out, so we can say that the bridge is really a * x(x-12), where we get to pick a to make sure that the height of the bridge matches what the question is asking for.
coarse ibex
To clarify, we could've also picked f(x)=a(x-6)(x+6), since that would zero out at -6 and +6, it's just a bit simpler to work with it starting at 0 instead of at -6.
elfin tangle
No LOL 
Can you write our your steps like one after another? So I see where the numbers are going etc?
coarse ibex
Well, you know that when you plug in x=6 you want the result to be 8, right?
elfin tangle
Hehe yes
coarse ibex
Can you write our your steps like one after another? So I see where the numbers ...
"Notice that we can pick the parabola f(x) = ax(x-12) so that it crosses the x axis at 0 and 12, giving us a bridge where the road has length 12. This is true for any value of a, so we can pick a suitable a so that our bridge has the right height. We know that at x=2 the height should be 8, so we can plug in f(2) = 8 and solve for a."
elfin tangle
How would that look?
coarse ibex
Solving for a?
elfin tangle
Mhm
coarse ibex
Well, what's f(2)?
coarse ibex
Well, yes lol
But you can also plug it into f(x) = ax(x-12)
You get an expression that involves a.
And then you know that expression also equals 8
Meaning you have an equation
elfin tangle
8=a(2)(2-12)
elfin tangle
a= -2/5?
coarse ibex
Yep
coarse ibex
And it matches what you got up here
elfin tangle
So what's the function that models the arch? Is that just it?
coarse ibex
Yeah, that's it, f(x) = -2/5 x (x-12)
elfin tangle
And how would I find the max?
coarse ibex
Well, where do you think the max is, at what x value?
Or rather, at what point along the bridge
elfin tangle
6
coarse ibex
Yeah, why do you think 6?
elfin tangle
Because 12/2 = 6
elfin tangle
Yess
coarse ibex
So now that we know where it is, can you say what the max height is?
coarse ibex
indeeeed
elfin tangle
14.4?
elfin tangle
I was eyeing this question for hours!!!
Thank you!!!!!!
Buttttttttt
I have another question
How the frigg would I do this LOL
coarse ibex
You want to find at what values of x are the graphs at the same height, since if they're at the same height at some x then they're intersecting. Now what does it mean for them to be at the same height at x? It means that f(x) = g(x). So basically you just need to solve $$-2x^2-7x+10 = -x+2$$
warm shaleBOT
Morrow
coarse ibex
Yeah
And then when you find x (there may be multiple x's) you can plug it into either function to figure out the y value. It doesn't matter which function since you're specifically looking for x's where the functions return the same result!
coarse ibex
Looks good
coarse ibex
It's a bit different. Here you have a parameter k and you need to figure out for what values of k do you get two solutions to the resulting equation.
coarse ibex
So you solve like normal, treating k like a plain old number until you get to the quadratic formula. Do you remember the criterion for when the quadratic formula gives you two solutions?
elfin tangle
You write both?
coarse ibex
Does the term "discriminant" ring a bell?
elfin tangle
A little LOL
elfin tangle
OH yes
coarse ibex
So what does the discriminant tell you about the number of solutions?
elfin tangle
No clue LOL
coarse ibex
Well, what happens when the discriminant is negative?
coarse ibex
Well, say you end up with, i dunno
$$\frac{-3 \pm \sqrt{-7}}{2}$$
what does that mean?
warm shaleBOT
Morrow
elfin tangle
it doesnt work\
elfin tangle
the solution is 0?
coarse ibex
Not necessarily. Going back to the above example, what happens if you have $$\frac{-3 \pm \sqrt{0}}{2}$$
warm shaleBOT
Morrow
elfin tangle
You get a decimal number
coarse ibex
Specifically, you only get one solution
elfin tangle
TRUE
coarse ibex
because +0 and -0 are the same thing
elfin tangle
yes
coarse ibex
So this leaves us with when the discriminant is positive. In that case you get two solutions, right?
elfin tangle
yep!
coarse ibex
Great. So now, here's the plan. Solve the equation, treating k like a plain old number, like as if it's 6 or something. Then when you get to the point where you'd apply the quadratic formula, just check for which values of k is the discriminant greater than 0.
elfin tangle
??
coarse ibex
You're trying to figure out when there are two solutions right?
coarse ibex
And we just said that happens when the discriminant is positive
So that's it, we figure out what the discriminant is, in terms of k, and then figure out for which values of k it's positive, i.e. greater than 0.
elfin tangle
I still dont get it LOL
coarse ibex
At what part do you stop following
Do you follow that when the discriminant is positive that means that there's two intersection points?
elfin tangle
Yes
coarse ibex
So how about you first figure out what the discriminant is?
coarse ibex
Do the exact same thing you did in the last question
coarse ibex
I have to input k as b?
k will end up as part of the c, since remember, we're just treating k like a plain old number, like 7.
elfin tangle
Im@still so confused 😭
coarse ibex
The first step is the same as the last question.
$$-x^2 + 4x + k = 8x - 2$$
warm shaleBOT
Morrow
coarse ibex
Can you get this equation into the form ax^2 + bx + c = 0?
elfin tangle
Mhm!
coarse ibex
So go ahead
elfin tangle
-x^2-4x+k+2=0
coarse ibex
Great, so in this case, what are our a, b, and c?
elfin tangle
1, -4, k+2?
coarse ibex
close, a is -1
elfin tangle
oops i forgot about the negative LOL
coarse ibex
So what's the discriminant?
coarse ibex
16 + 4(k+2)
16 + 4k + 8
24 + 4k
is what I got
Which is 4(k+6)
So when is 4(k+6) > 0 ?
Meaning for which values of k is that true?
elfin tangle
Which is 4(k+6)
Oh
elfin tangle
Meaning for which values of k is that true?
No clue LOL
coarse ibex
No clue LOL
Well, when is k + 6 positive? Since if k + 6 is positive then 4(k+6) will also be positive.
(and vice versa)
elfin tangle
I still have no clue 😭
Can you list them step by step so I can see hwere you got each number from?
coarse ibex
We want to know when 4(k+6) > 0, since 4(k+6) is the discriminant
Divide both sides of the inequality by 4
k + 6 > 0
Have you worked with inequalities before?
elfin tangle
Not really
coarse ibex
Well, let me put it this way. If k is 0 is k + 6 positive?
elfin tangle
Yes
coarse ibex
How about if k is -3
elfin tangle
Yes
coarse ibex
And if k is -5.5
elfin tangle
Yup
coarse ibex
What about -7?
elfin tangle
Nope
coarse ibex
And -6?
coarse ibex
Yeah, so 0 is not positive
Remember if the discriminant is exactly 0 we get one solution, not two.
So do you see the pattern here?
elfin tangle
I do!
coarse ibex
So for which values of k is the discriminant positive?
Or, rather, when is k + 6 positive?
coarse ibex
Well, yes, but not only when k is positive
for example we said that k = -3 also works
And so does k = -5.5
coarse ibex
To be clear here, the solution is k > -6
elfin tangle
I know!
coarse ibex
The steps are:
- Solve the equation up until the point where you'd use the quadratic formula.
- Calculate the determinant.
- Solve the inequality "determinant > 0" (solving for k)
For that last part, you'd use the laws for manipulating inequalities.
4(x+6) > 0 \\ multiply both sides by 1/4
x + 6 > 0 \\ add -6 to both sides
x > -6
elfin tangle
I get it nowww
coarse ibex
glad I could help
obtuse pebbleBOT
@elfin tangle Has your question been resolved?
timid silo
how do I calculate the ?
obtuse pebbleBOT
how do I calculate the **?**
royal basin
is the questionmark an angle?
timid silo
yes
royal basin
then there is not enough info
timid silo
wdym
royal basin
there is not enough information to calculate the angle you're asking for
royal basin
couldn't say, i don't know where this diagram came from
timid silo
couldn't say, i don't know where this diagram came from
I created it