FlexO
#help-41
grizzled pagodaBOT
red knot
Aura -
@swift shore
That ' means the function is differentiated
It's (√x) . (x² - √x)
You can interpret it as f(x).g(x) and solve accordingly
red knot
swift shore
Oh whatever guess it doesn’t matter 🤦♀️
My teacher didn’t teach me anything like that
swift shore
I mean I’ve just learnt this so I’m not like pro sorry 🤷♀️
I’ve been using dy/dx
But ty anyways for trying
.close
amber waspBOT
amber waspBOT
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golden pewter
Hi, can someone please help me with this induction problem? I tried substituting in the last equation using the hypothesis, but I was not able to prove it that way.
golden pewter
The problem does not directly suggest using MI, but I think that's the method I should use.
quick ridge
split the first k terms off from the sin(k+1…
$\frac{\sin(kx/2) \sin(\frac{k+1}{2} x)}{\sin(x/2)} + \sin((k+1)x)$
grizzled pagodaBOT
knief
golden pewter
Ok)
Should I multiply both sides by sin(x/2), or leave it as it is?
This equals the right hand side of P(k+1).
golden pewter
Yesterday I tried to combine terms in different ways like 20 times ... I was not able to get an equality here
or maybe we should use half angle formulas? or double angle
but then we get cos
golden pewter
sin2x? yeah
it turned into a huge mess lol
half angle formulas would have sqrt in it, so I didn't try that
golden pewter
expressing sin(kx/2)*sin((k+1)x/2) as a sum? Honestly, I don't remember, I can try that now.
but I would get sum of cosines, is that ok?
quick ridge
go with it
you should be able to get something
i don’t see how else you would do this
product to sum maybe
for the numerator
quick ridge
maybe don’t do too much angle sum
i’m sure it works
but
it seems like excessive
cant you use product to sum
golden pewter
Where?
quick ridge
2sin(u)sin(v) = cos(u-v) - cos(u+v)
golden pewter
Yeah, that's what I did for the first term)
quick ridge
we have $\frac{\sin(kx/2)\sin(\frac{k+1}{2}x + \sin(x/2) \sin((k+1)x)}{\sin(x/2)}$
oops
grizzled pagodaBOT
knief
we have $\frac{\sin(kx/2)\sin(\frac{k+1}{2}x + \sin(x/2) \sin((k+1)x)}{\sin(x/2)}$
```Compilation error:```! Argument of \trigbraces has an extra }.
<inserted text>
\par
l.49 ...}{2}x + \sin(x/2) \sin((k+1)x)}{\sin(x/2)}
$
I've run across a `}' that doesn't seem to match anything.
For example, `\def\a#1{...}' and `\a}' would produce
this error. If you simply proceed now, the `\par' that
I've just inserted will cause me to report a runaway
argument that might be the root of the problem. But if
your `}' was spurious, just type `2' and it will go away.```
golden pewter
maybe you forgot to close a { somewhere?
lol
I can see you making progress with latex tho! this equation is just too messy
grizzled pagodaBOT
knief
\[\frac{\sin(\frac{kx}{2}) \sin(\frac{k+1}{2}x + \sin(\frac{x}{2})\sin((k+1)x)}\]
```Compilation error:```! Argument of \trigbraces has an extra }.
<inserted text>
\par
l.49 ...+1}{2}x + \sin(\frac{x}{2})\sin((k+1)x)}\]
I've run across a `}' that doesn't seem to match anything.
For example, `\def\a#1{...}' and `\a}' would produce
this error. If you simply proceed now, the `\par' that
I've just inserted will cause me to report a runaway
argument that might be the root of the problem. But if
your `}' was spurious, just type `2' and it will go away.```
golden pewter
my cs homework must be completed using latex and it is due in 2 days ... I know NOTHING about its syntax, guess I'm cooked ...
icy solar
Hi, can someone please help me with this induction problem? I tried substituting...
a past question here: #geometry-and-trigonometry message
grizzled pagodaBOT
vin100
quick ridge
surely you can do this without that though
golden pewter
a past question here: https://discord.com/channels/268882317391429632/3261387574...
Sorry, but I don't understand the explanation there ... is it proven using induction?
icy solar
Sorry, but I don't understand the explanation there ... is it proven using induc...
not really
but there shld b a way to pretend using it
golden pewter
This question is from a problem set where I was able to prove the rest using MI, so I assume there must be some similar approach for this one too)
$[
\frac{\sin\left(\frac{kx}{2}\right) \sin\frac{(k+1)x}{2}\right + \sin\left(\frac{x}{2}\right) \sin\left((k+1)x\right)\right)}
]$
grizzled pagodaBOT
a5667.
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
icy solar
$\[
\frac{\sin\left(\frac{kx}{2}\right) \sin\frac{(k+1)x}{2}\right + \sin\left(\...
you're already in math mode inside $...$
you don't need an extra pair of backslash-escaped sqaure brackets \[...\]
grizzled pagodaBOT
vin100
icy solar
,,\int_a^a f(x) ,\dd{x} = 0.
grizzled pagodaBOT
vin100
icy solar
for \displaystyle math expressions
golden pewter
lol
Ok let me try again
so I'd have to type
,,\frac{\sin\left(\frac{kx}{2}\right) \sin\frac{(k+1)x}{2}\right + \sin\left(\frac{x}{2}\right) \sin\left((k+1)x\right)\right)}.
umm I think I messed up again
icy solar
lol
Ok let me try again
so I'd have to type
,,\frac{\sin\left(\frac{kx}{2}\r...
i suggest using latex workshop in vscode
for syntax highlighting
ofc you can also use markdown codeblock
\LaTeX{} code with $a = 1$ syntax highlighting
vin100
golden pewter
i suggest using latex workshop in vscode
I downloaded the extension but I cannot install miktex on win os for some reason
icy solar
what extra brace
you can cancel out the correct part to find out the incorrect part
i.e. remove the \frac{...}{...} inside
then the syntax error shld b obvious
icy solar
I downloaded the extension but I cannot install miktex on win os for some reaso...
i have a basic online math editor:
https://lstu.fr/live-math
it's possible to download it and use it offline
icy solar
provided that all the external links were changed to local ones
icy solar
overlead works fine
if you think that's the bot's problem, then you may report on @grizzled pagoda's official server
icy solar
I downloaded the extension but I cannot install miktex on win os for some reaso...
you may ask for support on
- TeX.SE
- #latex-help
- @grizzled pagoda's official server's LaTeX help
golden pewter
[
\sin\frac{kx}{2}\sin\frac{(k+1)x}{2} + \sin(kx + x)\sin\frac{x}{2}
]
grizzled pagodaBOT
a5667.
icy solar
**a5667.**
there's a slightly shorter syntax
golden pewter
you may ask for support on
1. TeX.SE
1. <#840667252793802752>
1. <@510789298321...
the file just doesn't get downloaded ... does it work for you?
icy solar
,,\sin\frac{kx}{2}\sin\frac{(k+1)x}{2} + \sin(kx + x)\sin\frac{x}{2}
grizzled pagodaBOT
vin100
golden pewter
oh ... yeah that's much better!)
golden pewter
the file just doesn't get downloaded ... does it work for you?
I think their website isn't working properly tbh
icy solar
the file just doesn't get downloaded ... does it work for you?
,tex I have installed Mik\TeX{} when I bought me computer
grizzled pagodaBOT
vin100
icy solar
my current 💻
grizzled pagodaBOT
vin100
golden pewter
https://miktex.org/download
if you click on download here, does it do anything for you?
I tried texlive too but it just keeps loading the file forever for some reason
also why do these doc pages look so outdated lol ... no UI whatsoever🫠 cuz why bother right
golden pewter
**a5667.**
@quick ridge what do you think I should do starting from here? basically from scratch
quick ridge
did you try product to sum
golden pewter
yeah we got this result
quick ridge
hmm id have to give it some thought but i have stats hw due at midnight
when i finish i can help
surely you dont need e^itheta though
golden pewter
oh, sure
this is not homework I just self study, I can come back to this anytime)
I will skip it for now
thank you so much for your help @quick ridge @icy solar
I will try to find some github copies of miktex (but it's a zip not .exe which makes things complicated) if the website keeps glitching))
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amber waspBOT
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noble coral
If I have to multiply 4 matrices together do I just first multiply the first 2 and then take that product and multiply it by the third and then take that product and multiply it by the fourth?
pallid canopy
Yes matrix multiplication is associative
amber waspBOT
@noble coral Has your question been resolved?
noble coral
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amber waspBOT
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Available help channel!
meager gale
amber waspBOT
meager gale
hi i highkey forgot how to do part b
amber waspBOT
@meager gale Has your question been resolved?
meager gale
<@&286206848099549185>
ivory pivot
oh what did you get for part (a) first
meager gale
b=-7
ivory pivot
I believe so
meager gale
ah so a = -4
ivory pivot
a(x-2) = a((x+1)-3)
meager gale
a(x-2) = a((x+1)-3)
i do this?
ivory pivot
how to get b
part b or the value of b?
ivory pivot
value of b
I don't believe that's correct tbh
meager gale
I don't believe that's correct tbh
wht else would it be?
cause -7 is just for divide by (x-2)
right?
ivory pivot
wht else would it be?
b is -7
no?
if we use the remainder thm
and we plug in p(2)
we get b = -7
meager gale
yea buts thats only for divide by 2
x-2
we finding for the whole thing
so it would be diff no?
ivory pivot
a and b are constants
meager gale
nvm
meager gale
a and b are constants
i see what you mean now
ohhh
hng on. sec
im cooking
a(x-2)
= -ax -2a
so
-ax -2a +b
= -4a +1
right?
meager gale
what are you trying to do
i was jsut thinking
cause like
b is a constant
but its not the constant term right?
cause P(x) has (x-2)
so theres nore than b
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amber waspBOT
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heady moss
heyyy i need help
heady moss
amber waspBOT
@heady moss Has your question been resolved?
floral patrol
when im solving this does bedmas matter between each fraction?
floral patrol
do i have to divide the last two fractions first or does the order not matter
worldly estuary
start by division
floral patrol
alr thanks
thats what i was doing but the answer sheet is off from mine
im gonna try again
.close
steep fern
hi my question is "is there a number that is exactly 1 more than its cube"
timid vault
You mean a number such that:
x³ + 1 = x?
steep fern
yes
steep fern
how would i prove it through an equation? do i plug in like random numbers until it works out
spare belfry
how would i prove it through an equation? do i plug in like random numbers until...
Just explain that a cubic function must have at least one real root
steep fern
because i tried subtracting x so i could make f(x) = x^3 + 1 -x
ohhh
tyy i think i understand it now
.close
unkempt imp
how to do this?
amber waspBOT
how to do this?
ivory sorrel
how to do this?
Would you agree that the first step would be to substitute for 2x?
unkempt imp
nope
trig identities
take out cos
but I dont know what to do from there
make u = cos is that right?
ivory sorrel
make u = cos is that right?
I'm not sure how helpful that would be
There is a standard method to solve "odd powers of sin and cos" which I know of
unkempt imp
that I dont know
unkempt imp
but I know like if function is odd let the U be the other function
but I just dont know which one is priority
ivory sorrel
amber waspBOT
@unkempt imp Has your question been resolved?
unkempt imp
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amber waspBOT
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Available help channel!
amber waspBOT
sullen tapir
There is a formal definition of if
If A then B is defined by a truth table
When A is true B is true
When A is false
Im gonna rip one off the internet
Yeah
That defines syntactic implication
4 cases total cause 2^2
Two combinations of two truth values
And yes
There is an idea in math known as explosion, we want to prove everything from false statements
So false implies anything
Its like how when proving something for algebra you end up with 3=2 and say “the thing i started with must be wrong”
So if the sky is red, then there will be a storm
We can disprove A->B by considering the case A and not B
Is it possible that the sky could be red but there cannot be a storm?
Does the red sky mean a storm will ALWAYS happen?
Like irl on real earth
Right
So its wrong
You picked right
A->B means that it is immpossible for A but not B
LOL
sullen tapir
A->B means that it is immpossible for A but not B
Does this make sense though? This is important
sullen tapir
Does the red sky mean a storm will ALWAYS happen?
Do you see how this connects to the truth table
Go ahead just ask any time
regal rivet
What did I do wrong?
amber waspBOT
What did I do wrong?
regal rivet
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amber waspBOT
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worn slate
I'm having a hard time with this equation. I've tried grouping it but it didn't work
amber waspBOT
I'm having a hard time with this equation. I've tried grouping it but it didn't ...
@worn slate Has your question been resolved?
amber waspBOT
amber waspBOT
Available help channel!
languid thorn
okay so
f(x)=x^3
g(x)=1/x
need to find fog
amber waspBOT
okay so
f(x)=x^3
g(x)=1/x
need to find fog
languid thorn
but i get that the answer is 1/x^3, i just dont get why when doing this u do like f(1/x) then u do f then it turns into 1/x^3 like what about the numerator
why does nothing happen to that
is f just the ^3?
half rover
but i get that the answer is 1/x^3, i just dont get why when doing this u do lik...
because 1^3 = 1?
amber waspBOT
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Available help channel!
tulip tapir
how might i solve this?
amber waspBOT
how might i solve this?
grizzled pagodaBOT
ren
tulip tapir
my mistake, it's this instead
grizzled pagodaBOT
ren
rancid minnow
What's is the chance a 1% event will happend 15 times in a row
cunning birch
if everytime it happens is independent from the other
then with a chance of p for the one-time event
repeating it n times will just make the probability p^n
rancid minnow
Imagine you have 1% to pass a gate and you need to pass 15 of them all have 1 precent
cunning birch
uh huh
rancid minnow
How many tries will it take
cunning birch
on average?
so chance of p = 1/100 to pass one gate
every gate pass is independent from the other
repeat n = 15 times
cunning birch
?
rancid minnow
Ah
rancid minnow
Lemme get calcalator
cunning birch
now say you repeat the process
until you finally get past all the gates
and X be the number of times it took you to pass all the gates
rancid minnow
.close
maiden stratus
is vsquare = u square + 2as only used when u dont have t for v = u + at or there are other reasons too?
amber waspBOT
is vsquare = u square + 2as only used when u dont have t for v = u + at or there...
@maiden stratus Has your question been resolved?
rancid raven
is vsquare = u square + 2as only used when u dont have t for v = u + at or there...
These things depends what parameter are given, if you have v,u and a given and you need distance ie S then you can use first
If you want time t, use second
amber waspBOT
maiden stratus
These things depends what parameter are given, if you have v,u and a given and y...
thanks
amber waspBOT
Available help channel!
thorny grotto
So i'm getting back into algebra after a year and I remember it but also don't. I got a problem that i'm trying to remember how to solve if some can help me with that
worldly estuary
Factor out the denominator
worldly estuary
No? Not sure what do you mean by this
A rational expression is undefined when the denominator is equal to zero
thorny grotto
Yeah, I'mma be completely honest. I understand math but not the terms so sorry. Saying factor out the denominator made little sense to me. Do I divide the 15 with y above?
worldly estuary
That's not how factoring works 
Did you write out y=y/15 there or it was given like that
In the box
worldly estuary
Factor out the denominator
What I mean by this is factoring the 3y+15
And what it becomes after we factor it
thorny grotto
Did you write out y=y/15 there or it was given like that
I did
and what are we factoring?
thorny grotto
what is factoring?
thorny grotto
Then it is wrong
I know
worldly estuary
what is factoring?
Google it
thorny grotto
So you want me to turn the y's into (3+y) and (4+y)?
worldly estuary
There's a gcf
thorny grotto
gcf?
worldly estuary
Greatest common factor
Factoring is usually finding the terms that are multiplied together to get an expression
worldly estuary
Not quite
For example, when you have 4x+8, gives us 4(x+2) since there is a gcf
Now do the same thing for yours
thorny grotto
3(y+5)?
worldly estuary
Yep!
thorny grotto
I never learned this before at least I don't think I have
worldly estuary
You can always learn in this server a lot dw
thorny grotto
Wait never mind
I think I remember now, yeah. but now I would divide the 3 to the top?
worldly estuary
A rational expression is undefined when the denominator is equal to zero
It is asking this
So set 3(y+5) equal to 0
thorny grotto
worldly estuary
So set 3(y+5) equal to 0
I saw but answer me this
thorny grotto
so put the bottom to 0?
thorny grotto
I'm unsure as well becasue it always say that and this
Doesn't say but guessing how I can put fractions. I think it would want both
worldly estuary
I think they meant to write either normal expression or fraction expression
Undefined is always the denominator
What happens if you put x=-5 only?
If the expression was 4y(x+5) then you could say 4y, y+5
Wait it is just y, ignore the x
thorny grotto
i'll try the 4y, y+5
worldly estuary
Okay
thorny grotto
Nope
worldly estuary
So set 3(y+5) equal to 0
Here
thorny grotto
Yeah that's fair. I'm cooked, this class is a self study
thorny grotto
Yeah, but I don't understand the teacher
worldly estuary
Teachers might be terrible at teaching but you can study on your own
I've heard a lot from this server and I think I'm good enough
amber waspBOT
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Available help channel!
austere timber
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austere timber
Question 4
amber waspBOT
@austere timber Has your question been resolved?
austere timber
<@966749964883615814> Has your question been resolved?
No
jaunty shale
,rccw
grizzled pagodaBOT
jaunty shale
!show
amber waspBOT
Show your work, and if possible, explain where you are stuck.
amber waspBOT
@austere timber Has your question been resolved?
sand kraken
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@sand kraken Has your question been resolved?
sand kraken
<@&286206848099549185> sorry, i just really dk how to approach this 😭 nothing i try works
answer is 1/112 btw
sinful sapphire
1/112 makes sense
like suppose the first driver drives the car
there's 8c3 options liek that
and 8c3 more if the first driver drives the van
so 1 times out of 112 they all do the same thing twice
sand kraken
why would it be 8c3 more if the first driver drives the van
sinful sapphire
🤷♂️
sinful sapphire
like "both in car" makes no sense
you;re probably solving something other than what they meant
they don;t mean 2 specific people
they mean all 10 at the same time do the same thing
sand kraken
i thought it meant there are two drivers, and they just wanna know probability of them both being in the car or both in the van
sinful sapphire
that would never happen
they would avoid it
so that there's someone who can drive in each vehicle
sand kraken
uhhhh ok wait i see what u mean
sinful sapphire
which yeah, the question keeps secret from you
sinful sapphire
it's badly worded
sand kraken
yeah omg you're right tysmm i was losing my mindddd 8C5 is the same as 8C3 so still 56+56
ty!
sinful sapphire
np
sand kraken
.close
celest thorn
amber waspBOT
celest thorn
How to compute the limit at n -> infinity
i applied binomial theorem and got 8 but it isnt correct
cunning birch
Oh no that's not how it works
Even if all the other terms go to 0 individually, eventually there's an infinity of them
Just like the sequence
1
1/2 + 1/2
1/3 + 1/3 + 1/3
...
Doesn't go to 0
celest thorn
oh okay
celest thorn
yes
celest thorn
like this?
i can take out n from the power and put it behind log
what else?
@cunning birch sorry for ping! can i get a hint for the next step please?
cunning birch
So its log[(1+7/n)^n]
cunning birch
And to be honest i prefer the form n*log(...)
So now only looking at the exponent
Write x = 1/n
Rewrite the exponent in terms of x
celest thorn
done
cunning birch
Alright so what does it give you
celest thorn
cunning birch
Yeah why not
cunning birch
Which is correct
hazy whale
ok so im just confused what to discontinuities to call a function
hazy whale
in my notes i have f(a) not existing means its removable discontinuity, Lim DNE means Infinity discontinuity, and jump discontinuity for f(a) =/ Lim
but some example problems say otherwise
in one Lim DNE and they said its jump discontinuity
pallid canopy
but some example problems say otherwise
show the examples
pallid canopy
what is "it"
hazy whale
the discontinuity at x = -3
pallid canopy
your notes are wrong
hazy whale
can u correct my notes o.o
pallid canopy
jump discontinuity is correct
hazy whale
so how do i correct the notes
amber waspBOT
@hazy whale Has your question been resolved?
amber waspBOT
@hazy whale Has your question been resolved?
wicked dust
hazy whale
wat
amber waspBOT
@hazy whale Has your question been resolved?
amber waspBOT
@hazy whale Has your question been resolved?
amber waspBOT
@hazy whale Has your question been resolved?
amber waspBOT
@hazy whale Has your question been resolved?
patent raptor
in my notes i have f(a) not existing means its removable discontinuity, Lim DNE ...
Lim DNE means Infinity discontinuity,
not necessarily, as you can see it can also be a jump discontinuity
If the limit is indefinite, we say it doesn't exist basically
patent raptor
From the left and right side as x->-3 you approach not the same value and thus the limit doesn't exist. But that doesn't make it an infinite discontinuity.
Basically if you have a jump discontinuity then the limit doesn't exist for sure, but the inversion is not necessarily true.
split sail
wondering if i'm doing this correctly, can someone please check my answer? Thank you!
lusty pine
how did you get your answer?
split sail
how did you get your answer?
basically
i started by simplifying the cubed root
wait lemme take a picture of my work
Hint, roots are fractional exponents (square root is equivalent to the 1/2 power)
Just a lil something to throw out there
split sail
Hint, roots are fractional exponents (square root is equivalent to the 1/2 power...
it said in simplified radical form so i thought of it to leave it as that
Looks messy asf but I tried
lusty pine
how did
$$\sqrt[3]{\sqrt{x^6}}$$
turn into
$$\sqrt[4]{x^6}$$
grizzled pagodaBOT
Skill_Issue
lusty pine
multiplying the radicals
what did you multiply
split sail
what did you multiply
the
OH SHOOT
it would be sixth root
right?
not wtv the heck that is
lusty pine
in here 6/4=2?
split sail
what my teacher said
is that you divide the index by the exponent
and if it goes 1 time you put the x out and keep the remainder in
lusty pine
err whats an index
split sail
basically the 4
lusty pine
oh sorry olg lmfao i didnt see that x
yeah thats correct
so sorry didnt see the x thats out of the sqrt lol
split sail
oh
lmo
but its uh technically still wrong???
cause it should be sixth root
not fourth
lusty pine
it would be sixth root
yea cus this
lusty pine
also, unless im misunderstanding (again) it shouldnt be x sqrt(x^2)
split sail
well
the index goes into the exponent 2 times
so i got that wrong probably
maybe it should just be x^2
split sail
actually just try to fix the root 6
how
lusty pine
also, unless im misunderstanding (again) it shouldnt be x sqrt(x^2)
btw here im p sure it should be $x\sqrt[4]{x^2}$
lusty pine
it would be sixth root
you said here it should be 6th root right instead of 4th root?
lusty pine
mhm
split sail
how
split sail
OH SHOOT
lusty pine
actually ill just delete that so no misunderstandings :D
lusty pine
so now you have x^2sqrt(x)/x
lusty pine
so now you have x^2sqrt(x)/x
simplify this and your golden
split sail
oh
THANK YOU SO MUCH
that would be x squareroot of x?
JUST to make sure (cause all hw in honors is graded 😭)
amber waspBOT
@split sail Has your question been resolved?
amber waspBOT
amber waspBOT
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ruby flower
quick factoring question
amber waspBOT
quick factoring question
ruby flower
the question is asking me to expand and simplify
and its structured like this
2(x-4)+5 (x+3)
so do i expand and then foil?
celest cove
only option, yeah
celest cove
yup
amber waspBOT
celest cove
you too
amber waspBOT
Available help channel!
civic spindle
1. How similar is this to traditional solving for X where you'd undo operations to solve for X?
2. Unlike solving for X in this context X can be any value and you solve for what input of X would generate zero given the numbers and operations applied giving the numbers precedence over X. This is why I dont understand how they can change the value by subtracting c/a as they appear to be analyzing the behavior of the problem including c/a rather than just X.```
civic spindle
Topic:
There appears to rather be a hidden object (goal) of their operations which uniquely ignores such operations (like how when solving for X you can undo everything to solve for it)
or
perhaps my understanding is fundamentally flawed regarding how the operations interact with the whole which would probably take a more flow-of-logic based approach to fix```
bronze basalt
there is not a single occurance of x, so it's not as simple as just undoing some operations
civic spindle
note: I like speaking very formatted as I enjoy doing so as it increases professionalism and reduces toxic social contact.
civic spindle
there is not a single occurance of x, so it's not as simple as just undoing some...
'Q:' How does anything regarding X influence the problem if they are focusing on the results of the quadratic operations performed such as C/A? Would it not generate different roots if such operations were changed?```The values of (x) that satisfy [x^2 + \frac{b}{a} x + \frac{c}{a} = 0]
and the values of (x) that satisfy [x^2 + \frac{b}{a} x = - \frac{c}{a}]
are exactly the same.
grizzled pagodaBOT
Invariance
civic spindle
you're really good at wording things
civic spindle
**Invariance**
thinking:
I am confused regarding how exactly this is the case, if the sign became negative how would the input value stay at zero? Because logically, if prior it added by C and that made it go up thrice then you found a value that equated to zero even after being added thrice then how come switching it to negative would still equate to zero? would that not result in a negative value?
civic spindle
We are looking for any value of X that satisfies the equation which I assume can be any value we want so long as it works out.
bronze basalt
yes, we start off looking for roots of a quadratic, but we rephrase that in equational form
For example
[(1)^2 + \frac{4}{2} (1) - \frac{6}{2} = 0]
and so
[(1)^2 + \frac{4}{2} (1) = \frac{6}{2}]
grizzled pagodaBOT
Invariance
bronze basalt
sorry fixed the signs there
civic spindle
we subtracted c/a from both sides
Given the only concepts you understood were that you were trying to find a value of X that generated zero after the operations given and that what matters is the behavior derived from said operations (get the roots that describe it)
and given the understanding that the goal is to graph the behavior of the operations **themselves** after finding an X that is at zero
this would then logically conclude that by ANY value becoming unequal the original behavior that we wanted to describe being lost would then be a different behavior and just losing it
how is it that these "roots" are the same despite everything? If you graphed this would everything still be the same?```But, based on all prior situations, I believe there is a very fundamental concept missing from my understanding that would explain why you can simply remove things from the equation while keeping the roots so long as you did it to both sides.
My understanding of subtracting from both sides is that the right side is currently equal to the overall result of all operations so by placing -c/a there it would be stored there until other operations were performed leading to the result when such other operations also subtract with -c/a
another understanding is that you are solving for whatever value is on the other side of the = where -c/a then was placed
which of these is more correct?```
You may be able to figure out what is missing by just thinking about what I did not mention as that is my full understanding.
bronze basalt
The expression (a x^2 + b x + c) is a polynomial with two roots. The equation (a x^2 + b x + c = 0) holds exactly when (x) is equal to one of those roots. Oftentimes people conflate expressions with equations because it's convenient, but I feel it's useful to draw a clear distinction for the purposes of this conversation
grizzled pagodaBOT
Invariance
bronze basalt
The roots of (a x^2 + b x + c) are given by (a x^2 + b x + c = 0). When (a \neq 0), the solutions to (a x^2 + b x + c = 0) and (x^2 + \frac{b}{a} x + \frac{c}{a} = 0) are identical (divide both sides by (a)). Similarly, the solutions to (x^2 + \frac{b}{a} x + \frac{c}{a} = 0) and (x^2 + \frac{b}{a} x = - \frac{c}{a}) are identical (subtract ( \frac{c}{a}) from both sides)). Therefore, the roots of (a x^2 + b x + c) are given by (x^2 + \frac{b}{a} x = - \frac{c}{a}) (when (a \neq 0))
grizzled pagodaBOT
Invariance
bronze basalt
```lua
My understanding of subtracting from both sides is that the right side is...
an equation is simply a statement that two things are equal
the equations (x^2 + \frac{b}{a} x + \frac{c}{a} = 0) and (x^2 + \frac{b}{a} x = - \frac{c}{a}) are identical in meaning (in the sense that one is true exactly when the other is true)
grizzled pagodaBOT
Invariance
bronze basalt
that's all there is to it
civic spindle
I think I see the part which may have been unexplained this entire time.
civic spindle
the equations \(x^2 + \frac{b}{a} x + \frac{c}{a} = 0\) and \(x^2 + \frac{b}{a}...
this is interesting... thinking:``` ```lua
But regarding what you said you said the solutions are identical but you didnt really seem to explain why they were equal as those operations do appear as if the solutions would change and i still dont understand how it is they wouldnt```if (x = y), then (x - k = y - k) for any (k), and vice versa
grizzled pagodaBOT
Invariance
bronze basalt
do you agree with that?
civic spindle
regarding equations is there some type of special thing going on where equality of some type of goal or operation takes precedence over the actual constituents leading to ANY value (delete c/x divide c/x remove b/a) being equal so long as some criteria is met?```Mathematically, there is no additional meaning on equations other than asserting that two things are equal. "goals" or "operations" don't play any direct role
civic spindle
**Invariance**
If x=y because it is x and you subtract a random ammount from x how is it still equal to y? Is the value somehow locked?```How can this be the case? Isnt the goal of a quadratic to describe the behavior of the quadratic rather than X? Of course using the zeroes to find the beginning of such operations (zero).
civic spindle
**Invariance**
im just blind i didnt see -k after y
bronze basalt
we may have goals we want to accomplish with the equations, but that's different from the equations somehow having those goals themselves
civic spindle
**Invariance**
yes
bronze basalt
**Invariance**
but this
bronze basalt
**Invariance**
and this
are the same thing
in both cases, we subtract something from both sides of the equation
to be clear, I mean [x = y \implies x - k = y - k]
and [x^2 + \frac{b}{a} x + \frac{c}{a} = 0 \implies x^2 + \frac{b}{a} x = - \frac{c}{a}]
are the same
grizzled pagodaBOT
Invariance
bronze basalt
(the arrow means "if the thing on the left is true, the thing on the right must also be true")
civic spindle
**Invariance**
but isnt the bottem one x = x and the right one y - k? where is the subtraction?
what exactly does = 0 mean?
my understanding of it was that the answer was not determined yet because the operations were incomplete
so you store things there while you are doing things (tracks results)
bronze basalt
no, it just means that x^2 + b/a x + c/a is zero
like if you have values for x, a, b, c, then putting x^2 + b/a x + c/a into a calculator should give a result of 0
civic spindle
reallllly think its as i thought and theres just some type of overarching goal or concept that negates all affects in pursuit of a goal.
civic spindle
like if you have values for x, a, b, c, then putting x^2 + b/a x + c/a into a ca...
btw off topic but how much harder is calculus than this
bronze basalt
Than algebra? It's... somewhat harder? The concepts are more complicated
bronze basalt
the stuff you need for CS, I wouldn't say so
civic spindle
im still just thinking about it (how its equal and the concept of equivalency)
amber waspBOT
Available help channel!
civic spindle
a
amber waspBOT
Available help channel!
split sail
How do you do #11, answer: 50 \sqrt{3}
amber waspBOT
How do you do #11, answer: 50 \sqrt{3}
@split sail Has your question been resolved?
chilly jackal
Have u been taught the law of sines?
split sail
Yes
raven glen
Yes
Here, since you know the angles and lenght of wiper, you can find x and y, the sum of which should be equal to the horizontl distance covered by the wiper
split sail
Oh ok thanks for the help
amber waspBOT
@split sail Has your question been resolved?
amber waspBOT
amber waspBOT
Available help channel!
keen pawn
amber waspBOT
robust isle
you didn't include Q and R...
keen pawn
Okay, so I CAN't do this, can I.
robust isle
this isn't a complete truth table
amber waspBOT
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Available help channel!
solemn escarp
how to solve the indefinite integral of sin^4 x?
amber waspBOT
how to solve the indefinite integral of sin^4 x?
native stag
I think I have an idea.
solemn escarp
?
outer abyss
how to solve the indefinite integral of sin^4 x?
Rewrite as sin^2 x (1 - cos^2 x)
solemn escarp
oh
split sail
OMG integration OMG
solemn escarp
i havent learnt u sub yet
native stag
Split it into two
oh lmao
outer abyss
i havent learnt u sub yet
You don’t need it here
outer abyss
Well ig you kinda do but it’s only u = 2x
solemn escarp
?
solemn escarp
sin^2 and sin^2 cos^2
native stag
you know $\int \sin(x)^2$ dx?
solemn escarp
yeah
grizzled pagodaBOT
Percy
native stag
the $(\sin(x) \cos(x))^2$ should look familiar.
solemn escarp
sin2a
grizzled pagodaBOT
Percy
native stag
write stuff properly lmaoo
solemn escarp
i got 7/32 (2x - sin2x)
native stag
for the entire integral?
solemn escarp
yeah
native stag
it doesn't seem very correct
solemn escarp
oh
native stag
its very much not
solemn escarp
wait what am i supposed to do with sin^2 (2x) then?
native stag
sub 2x as u
civic spindle
1. How does algebra fundamentally work in regards to moving things pass the equal sign while maintaining equality? (give examples)
2. I don't understand how they removed c/a and made it negative while maintaining the same root.```
civic spindle
civic spindle
Example someone gave:
split sail
One of Euclid's axiom says that subtracting or adding the same thing to the same thing will maintain equality.
civic spindle
One of Euclid's axiom says that subtracting or adding the same thing to the same...
9 = 9 | 9 - 9 = 0
context? That doesnt really appear to work here at least not within "equality."
unless you used the word in a uhm... unfortunate manner.
split sail
What I meant was that it will maintain the truth of the equation.
Given that the previous equation was true in regards to the situation, of course.
haughty token
9 = 9 | 9 - 9 = 0
what exactly are you trying to show here?
civic spindle
so in the sense that 9-9 = 0 would still be true?
split sail
True.
One of Euclid's axiom says that subtracting or adding the same thing to the same thing will maintain equality.
civic spindle
are they the same
civic spindle
then 9 - 9 = 0?
split sail
They are the same equality; one is just more simplified than the other and both represent the same relation.
Between 'x' and 'y'.
civic spindle
see this the problem right
you guys talk like this to people who think equal works in the traditional sense
oh well
split sail
Well I didn't tell that x =y, and x-k = y-k
then x = x-k
What is meant by equality is that both sides of the = sign are equal.
x is equal to y and x-k is equal to y-k (which can be concluded from x being equal to y)
civic spindle
and are both K the same value?
civic spindle
so if i am getting this correctly
you think that
9=9 and 9-5=3 are equal?```or only 9-9=0 because it used itself to generate zero?
split sail
Nine is equal to Nine and 'Nine minus Five is equal to Four'.
No
Wait
9-5 = 4
bruh you tricked me
civic spindle
alright that is a weird definition of equal
craggy sundial
The truthiness of one, implies the truthiness of the other
so the truthiness of both equations are the same
civic spindle
because they still equal themselves?
craggy sundial
assuming you mean 9-5=4
civic spindle
because 9=9 and 9-5=4 are true?
split sail
yeah
craggy sundial
yes, both are true
split sail
both are true statements
split sail
Wdym
split sail
?
civic spindle
civic spindle
they said the roots wouldnt change even if c/a was removed
is this also true for b/a
and if so what about X?
craggy sundial
the roots are the values x can be
civic spindle
well yes
split sail
𝒜(Arpeture)
civic spindle
but what if you removed b/a and c/a
craggy sundial
depends how you remove them
civic spindle
𝒜(Arpeture)
why did you call me by my other name xd
civic spindle
depends how you remove them
in this case i dont understand because they subtracted positive c/a and put negative c/a on the other side of the equal sign
if this were done using fundamentalist logic
then
civic spindle
if this were done using fundamentalist logic
well idk how exactly to show it the closest i can get is
craggy sundial
because they hid a step
civic spindle
5+5=X```they subtracted c/a on both sides
civic spindle
5=X-5```mhm
craggy sundial
```lua
5+5=X```
what's the value of x on that one?
split sail
10
civic spindle
well if 5+5 = X then it is 10
craggy sundial
ok, so now, replace x with 10 on the second equation
is that second equation true?
civic spindle
i can see how subtracting 5 would still be X but reduced
civic spindle
is that second equation true?
oh wait
craggy sundial
well, that's a different equation
pick values of a, b, and c to make this a bit more simpler
civic spindle
ok so
craggy sundial
then graph it, and find for what x values the equation before they "removed" the c/a is true
craggy sundial
you can do that too
craggy sundial
```lua
15^2 + 5/a 15 + 10/a = 0```
for what values of a is that equation true?
civic spindle
civic spindle
for what values of a is that equation true?
thinking:
craggy sundial
just replace a with x ig
I was asking you to pick a, b, and c
instead you picked b, c, and x
btw
craggy sundial
nice, now graph it
craggy sundial
lemme try to graph it
oh I see the problem
there's no real solutions for that one
craggy sundial
lemme pick the a, b, and c
craggy sundial
it's because the parabola doesn't hit 0
civic spindle
wait so it does make a parabola when you graph it?
i thought someone said quadratics in this case werent parabolas
or that those were like separate
craggy sundial
yea, a quadratic does make a parabola when you graph it
a quadratic equation though, doesn't
let's use
x^2 + (5/2) x - 10/2 = 0
craggy sundial
yea, those lines are where the equation is true
civic spindle
are there two lines because there are two X?
craggy sundial
because we're not using y, it defaults to all y values
craggy sundial
are there two lines because there are two X?
yup, there's two x values where the equation is true
craggy sundial
yea
civic spindle
thinking:
craggy sundial
hover over one of the points on the x line
craggy sundial
try replacing the x's on the second equation with one of those values
just on the left hand side one
then calculate the result
craggy sundial
I meant the x^2 + (5/2) x
craggy sundial
by second equation do you mean
and yea
civic spindle
both x^2 and x?
craggy sundial
mhm
craggy sundial
your superscript dropped there
just type a 2
and remove that 2 on the right of it
craggy sundial
remove the = 10/2 for now
craggy sundial
lemme do it
see what it equals to?
very close to what's on the right hand side
10/2
the reason why it doesn't exactly equal is because that -3.81174 is only an approximation
craggy sundial
because the operations that we picked,
subtracting on both sides, transforms the equation to another equation that's also true if that original equation is true