What triangle?
#help-19
vernal yacht
fluid tundra
for this particular question yes this is true
vernal yacht
i kinda get it
You’re literally on the path of her approach
Great start btw, could you draw the partial graph of cos(1/2x) ?
silk crescent
What triangle?
on the unit circle
vernal yacht
on the unit circle
Oh, you’re using the unit circle. I thought you were trying to use Mqnic’s method
silk crescent
i have never heard of that method before in my life 😭
vernal yacht
i have never heard of that method before in my life 😭
Do you know what does cos function looks like?
silk crescent
this but since the range is halved now we’re only looking at the top triangle im guessing
silk crescent
Do you know what does cos function looks like?
ya
vernal yacht
this but since the range is halved now we’re only looking at the top triangle im...
@ivory grove Is this your suggested method?
lone bison
this but since the range is halved now we’re only looking at the top triangle im...
does acos(1/4) even have a nice form lmao
ok wait that prob doesnt matter
for this if you do z = 1/2 * x you also have to move the interval from the "x world" into the "z world". so the new interval would be 0 ≤ z ≤ pi
silk crescent
do u have a strategy that doesnt involve introducing new variables?
ivory grove
<@1356757542537265403> Is this your suggested method?
yeah that was what I was thinking, but the period thing works too
vernal yacht
do u have a strategy that doesnt involve introducing new variables?
z is just 1/2x
Nothing special
lean yew
do u have a strategy that doesnt involve introducing new variables?
graph sketching
vernal yacht
yeah that was what I was thinking, but the period thing works too
yeah, prob you should continue explaining. Cuz I have no idea what the heck is this
Most people here suggest sketching.
lone bison
i sometimes think its helpful to make new variables so you can just sort of convert the problem as a simpler problem
ivory grove
yeah, prob you should continue explaining. Cuz I have no idea what the heck is t...
idk what yall are talking abt atm tho
lone bison
instead of worrying about the 1/2 factor on x just bundle it into a new variable and deal with it later
silk crescent
for this if you do `z = 1/2 * x` you also have to move the interval from the "x ...
so basically multiply everything by 1/2
lone bison
solving cos(z) = 1/4on [0, pi] is easier
lean yew
substitute significant values of x into the function to see
silk crescent
i sometimes think its helpful to make new variables so you can just sort of conv...
yeah i see why people like doing that but for me it doesnt really help much lol
lean yew
for example, what would you get for x = pi/4? pi/2? pi? 3pi/2? 2pi?
that should give you an idea
lone bison
for example, what would you get for x = pi/4? pi/2? pi? 3pi/2? 2pi?
the thing is the constant factor on x makes some of these less nice
vernal yacht
Alr alr, we have to come to a consensus.
Should we continue with OP’s work or sketching cosine function? Things getting messy rn
lone bison
like cos(pi/4) is a nice value but if you try to sub pi/4 into x on cos(x/2) you'll really end up calculating cos(pi/8)
silk crescent
Alr alr, we have to come to a consensus.
Should we continue with OP’s work or ...
yeah all these different methods are confusing me 🥀 🥀
lean yew
i'll go with the majority here
vernal yacht
yeah all these different methods are confusing me 🥀 🥀
What do you think? You have the say basically
lone bison
yeah drawing the cos function might work since you dont need to actually solve it (i think)
silk crescent
this but since the range is halved now we’re only looking at the top triangle im...
well this is how i imagine it rn
is that fine? okay? wrong?
vernal yacht
We’re not going anywhere with everybody using different methods
vernal yacht
is that fine? okay? wrong?
Sure
ivory grove
To clarify
your strategy is to see what points satisfy cos(theta) on the unit circle, right?
ivory grove
this but since the range is halved now we’re only looking at the top triangle im...
I'm assuming that is what this is about
lone bison
its "how many times do these two graphs intersect within the blue area"
ivory grove
its "how many times do these two graphs intersect within the blue area"
you can talk about this later; we are doing the unit circle method
vernal yacht
its "how many times do these two graphs intersect within the blue area"
Nah, drop it. We’ll continue with OP’s work first
lone bison
hm ok
silk crescent
my strategy is imagining that the variable affecting x (1/2, in the case of cos1/2x) is basically affecting the range, so for this question we would only look the top half of the unit circle because 2pi/2=1pi. therefore, only one solution
i just wanna know if thats a valid way of thinking about it or if its situational
ivory grove
mhm, that seems like the right way to think about it
because right now you're told that x<2pi
if x<4pi for example
x/2 would be able to hit all the values between 0 and 2pi
and therefore your strat won't work
@silk crescent does that make sense?
silk crescent
wait u said im right but then said my strat wont work
silk crescent
x/2 would be able to hit all the values between 0 and 2pi
but yeah thats what i mean
lone bison
they said it wouldnt work if the question was asking for solutions in a different interval
lean yew
but your x < 2pi, so you can use it
ivory grove
depends on what the value of x can be
silk crescent
damn
lone bison
if the question was asking for x in 0 <= x <= 4pi then x/2 could be up to 2pi
silk crescent
then i still dont understand what 1/2 is supposed to mean 💔
ivory grove
then i still dont understand what 1/2 is supposed to mean 💔
0≤x<2pi means 0≤x/2<pi
which means right now your strategy works
if it were 0≤x<4pi, then that means 0≤x/2<2pi
which means you would have to think more carefully
it might not be only one solution
lean yew
then i still dont understand what 1/2 is supposed to mean 💔
you can think of it like this
you have 0 < x < 2pi, right?
so if you are now considering half of x
the entire range is halved
so you can divide the lower and upper bound by 2
silk crescent
which means you would have to think more carefully
wouldnt it just be two solutions then, considering im just looking at one full rotation?
lean yew
0/2 changes nothing
2pi/2 = pi
ivory grove
wouldnt it just be two solutions then, considering im just looking at one full r...
yes, IF x<4pi, which is not the case in this problem
I'm talking abt this just in case you see similar problems in the future
silk crescent
alright
lean yew
(aka pattern spotting)
ivory grove
alright
do you have any more questions?
if not, then exp and the others can tell you the graphing method
silk crescent
so you can divide the lower and upper bound by 2
to clarify
if the range given is x<2pi and its cos1/2x i can just make the range x<pi
but if i were given a range like
x<6pi
BASED ON WHAT I KNOW
lean yew
mhm
silk crescent
if it was still cos1/2x the range would then turn into x<3pi
lean yew
correct!
but don't forget that your lower bound shrinks too
for example
if your range was -2pi < x < 2pi
ivory grove
You’re gonna panic with your triangle stuff
no, not necessarily
silk crescent
You’re gonna panic with your triangle stuff
i doubt my teacher would be insane enough to actually give me a range like that 💔
lean yew
similarly
lone bison
im not saying you have to but this is why i think its helpful to introduce new variables sometimes
silk crescent
halving x gives you this range
-pi < x/2 < pi
alralr then i get it now
lean yew
if you were given 3x, for example
vernal yacht
no, not necessarily
Nope, he will. 200%
When other elements are involved, he’ll literally perish
ivory grove
i doubt my teacher would be insane enough to actually give me a range like that ...
-# he/she will, because that's what always happens
ivory grove
Nope, he will. 200%
When other elements are involved, he’ll literally perish
stop being pessimistic, it's the same idea
lean yew
a range of
-2pi < x < 2pi
becomes
-6pi < 3x < 6pi
because the entire range has been expanded by 3 times
silk crescent
-# he/she will, because that's what always happens
i already know how to find general solution and the angles so it should be fine, will take a long time, and might drive me insane, but should be fine
lean yew
if in doubt, just remember one thing
silk crescent
a range of
-2pi < x < 2pi
becomes
-6pi < 3x < 6pi
makes sense
lean yew
to find the new range of the function with a coefficient on the x
multiply the lower and upper bounds (left and right of the x) by that coefficient
if it's 1/2 x, then multiply lower and upper bounds by 1/2
if it's 3x, multiply lower and upper bounds by 3
vernal yacht
stop being pessimistic, it's the same idea
Not at all. Sketching the graph is the simplest.
ivory grove
Not at all. Sketching the graph is the simplest.
true
lone bison
in the original problem if you think of z = 1/2 * x and the original interval was 0 <= x <= 2pi then the z variable you defined tells you how to transform the interval
namely 1/2 * 0 = 0, and 1/2 * 2pi = pi
so the new interval becomes 0 <= z <= pi
lean yew
Not at all. Sketching the graph is the simplest.
that's what i usually do when this kind of problem shows up
because it then also allows me to deal with cos (x + some other angle)
lone bison
the problem was just asking for how many solutions right? and not the solutions themselves
in that case yeah drawing might work
silk crescent
alrighty
sounds good
probably gonna use these help channels again later today for other questions….
but anyways
.close
odd edgeBOT
vernal yacht
probably gonna use these help channels again later today for other questions….
Aren’t you curious about how the sketching method work?
lean yew
i'll jump in if i'm around
odd edgeBOT
✅
lean yew
what
vernal yacht
vernal yacht
oh sketching
Thought you need helpful points or sth
lone bison
desmos time
lean yew
oh if you have new points for me i'm down to listen
ivory grove
desmos time
-# geogebra better >:) /jk
lean yew
so do you know the general form of the cosine graph?
lean yew
so it looks like this, right?
silk crescent
yes
lean yew
yes i suck at mouse drawing pls no criticize :<
ok so
when you are told something like cos (x/2)
if you dw to stretch the graph
silk crescent
oh i see where this is going
silk crescent
yeah
lean yew
divide all of them by 2
silk crescent
yea
lone bison
notice that cos(1/2*x) loops every 4pi not 2pi
silk crescent
i think i know what ur tryna do tho
lean yew
oh yea sorry sorry
silk crescent
1/2 transforms the graph by stretching it out
lean yew
correct
silk crescent
therefore
lean yew
but instead of redrawing the graph
silk crescent
the frequency decreases
silk crescent
ohh
lean yew
cuz doing that gives you this new graph
lone bison
bc 4pi after the transformation gets mapped to 2pi before the transformation
lean yew
lone bison
thats what divides x by 2
lean yew
it's the same as halving its frequency, correct
lone bison
yes
lean yew
so instead of stretching / compressing graphs
silk crescent
now the graph only goes through the x axis once within the range of 1<x<2pi
lean yew
you can instead multiply the x-values by the reciprocal of the coefficient
lean yew
now the graph only goes through the x axis once within the range of 1<x<2pi
yes!
0* btw
not 1
lean yew
so if you have 1/2 x
you multiply the x-values on the graph by 2
if you have 2x, multiply the x-values on the graph by 1/2
you can think of it this way
the x-values are meant to mark the period of the graph, right?
and frequency is 1/period
that's why you multiply by the reciprocal of your x coefficient
silk crescent
okok
lean yew
yep, so graph sketching in a nutshell
oh
and this also allows you to see what happens if you have cos (x + some angle) also
if the angle added is positive, you shift the graph to the right, or shift the x-values to the left
i prefer to do the latter
because then you just need to add -(angle) to all of the x-values
vice versa for negative angles
lean yew
yep!
with this
you can even do stuff like cos(ax + y)
apply the coefficient shift from a first
then do + y afterwards
silk crescent
so basically
so transformations on the graph accordingly
based on the question
then see where the graph touches x axis
to find solutions
lean yew
yup!
silk crescent
nicee
lean yew
these are all transformations of trig graphs
there are technically two more transformations but i think you know them already
y cos x simply makes the amplitude y instead of 1
and cos(x) + y shifts the graph upwards by y
or downwards if y is negative
that's it
the four possible transformations of sin/cos graphs
lean yew
be our guest
lean yew
i'm super tempted to do difference of two squares here
but i don't think it applies
cuz if you can
lean yew
you can split it into (sin^2 + cos^2)(sin^2 - cos^2)
then the first () disappears because of trig identity
silk crescent
oh yeah nvm i see
lean yew
not sure if this is legal
ivory grove
not sure if this is legal
it seems to be
lean yew
reverse the positions of those two by sticking a negative out front
-(cos^2 - sin^2)
then i think you know what the last step is
silk crescent
i lowkey forgot difference of squares existed hehe…..
ivory grove
then i think you know what the last step is
@silk crescent do you see what this is equal to
silk crescent
yeah i do
the reason i didnt realize is cuz i legit forgot i could do difference of squares
initially
ivory grove
silk crescent
ONE LAST QUESTION BEFORE I GIVE UP THIS CHANNEL
anyways uhh for this one not sure what to do
cuz
i havent seen any trig identities similar to this
lone bison
note: sin^2(x) + cos^2(x) = 1
lean yew
ONE LAST QUESTION BEFORE I GIVE UP THIS CHANNEL
whenever things like this come up i always think of using 2 sin a cos a = sin 2a
silk crescent
whenever things like this come up i always think of using 2 sin a cos a = sin 2a
theres no 2 in front tho
lean yew
is there now
if there is none in front
just move the 2 to the other side of the formula
:>
lone bison
ONE LAST QUESTION BEFORE I GIVE UP THIS CHANNEL
if you rewrite the 1 as sin^2(x)/sin^2(x) then the fraction becomes (sin^2(x) + cos^2(x))/sin^2(x) = 1/sin^2(x) = csc^2(x)
lean yew
formulas are not fixed in stone. you can manipulate them for different forms
lone bison
oh wait wrong question
silk crescent
its numer 8, just to clarify
lean yew
from 2 sin a cos a = sin 2a
silk crescent
number
lean yew
you know that you have only sin a cos a
so why not kick the 2 over to the RHS of the formula
then it leaves you with what you want
silk crescent
rhs?
lean yew
right hand side
silk crescent
where am i taking a 2 from tho
silk crescent
yeah
lean yew
if i divide both sides by 2
sin a cos a = (sin 2a) / 2
and what's left on the left hand side?
the expression in your question
the rest should be pretty straightforward from here
silk crescent
never knew i could use identities like that
lean yew
oh definitely!
identities are not set in stone
for example right
you know speed is distance over time?
v = d/t
if you do know that
maybe you also knew that distance = speed x time?
silk crescent
yeah
lean yew
this is just a rearranging of the formula
v = d/t
multiply by t on both sides
vt = d
identities can be manipulated algebraically just like any other expression or equation
lone bison
ONE LAST QUESTION BEFORE I GIVE UP THIS CHANNEL
sin(2a) = 2*sin(a)*cos(a)
1/2 * sin(2a) = sin(a)*cos(a)
set a = 1/3 x
1/2 * sin(2/3*x) = sin(1/3*x)*cos(1/3*x)
silk crescent
wait so can i do this
lone bison
no
lean yew
nonono
you can't divide the angle by 2
you must divide the whole sine by 2
so you don't get sin x
you get (sin 2x)/2
you get this
silk crescent
alralr
lone bison
you can consider the equivalent identity
$\frac{\sin(2z)}{2} = \sin(z)\cos(z)$
clever fjordBOT
exp(2) = 7.38905609893065022723…
lone bison
sometimes doing a change of variables makes it easier to spot patterns
silk crescent
oh i get it now
got the answer
alright thats the last of my questions for now finally 🥀
time to let this help channel
rest
.close
odd edgeBOT
odd edgeBOT
Available help channel!
frosty sundial
odd edgeBOT
frosty sundial
sorry for messi handwriting
pastel orbit
anything you've tried so far?
pastel orbit
the x in the numerator makes this a lot easier, actually
trig sub won't be necessary :p
frosty sundial
u sub?
strange aspen
but idk how to do trig sub
U sub :3
pastel orbit
yup
strange aspen
is it sqrt(1-x^2)
Hint d/dx (x^2)= 2x
frosty sundial
oh shoot
i got the question right
btw it was ibp question i was just solving integral of vdu part
.close
frosty sundial
odd edgeBOT
latent scaffold
Split the fraction in two.
nimble blaze
technically yes, but overthinking
latent scaffold
The antiderivative of 1/x^3 is not ln|x|.
latent scaffold
No, one just needs to use the power rule.
frosty sundial
1/x^3 and -1/x^2
thats what it turned into
frosty sundial
No, one just needs to use the power rule.
oh shoot
im selling
this is what it shoulda been
.close
odd edgeBOT
odd edgeBOT
Available help channel!
silk crescent
not sure what identities i can use for this question thats kinda it
nimble blaze
try starting with factorisation
woven marlin
not sure what identities i can use for this question thats kinda it
Divide the left side of the equation by 2 and then use the first memorable identity.I don't know if this idea can be applied
silk crescent
try starting with factorisation
by substituting sinx with a variable?
because if thats what u mean i dont think thats possible
nimble blaze
a sub isn't required,
but it may help if you aren't comfortable working with the sin(x) present
nimble blaze
i'm suggesting factorisation because it is definitely possible
silk crescent
i dont see any number i can plug in so the equation equals zero though, which i think is what ur meant to do
when ur factoring larger polynomials
amber veldt
if you have
nimble blaze
what numbers are you trying?
amber veldt
Az²+Bz+C=0
silk crescent
what numbers are you trying?
just take one look at the equation
amber veldt
you look for numbers that add to B and multiply to AC
nimble blaze
what numbers are you trying
amber veldt
but we cant replace A with anything to make the equation equal to zero
its already equal to 0 in the problem!
woven marlin
8A^4 + 2A^2 - 1
Take 2 as a factor,and we have 2(4A^4+A^2+-1/2)
nimble blaze
numbers aren't limited to integers
amber veldt
am i missing something
maybe you forgot how to factor non-monic quadratics?
silk crescent
all i remember is
find an integer u can replace x with
where the equation equals zero
amber schooner
what
amber veldt
you look for numbers that add to B and multiply to AC
did you ignore this
amber schooner
find an integer u can replace x with
where is this coming from
amber veldt
factoring by grouping ring a bell?
silk crescent
i thought its different since the leading degree is 4
nimble blaze
oversimplification of rrt
roots aren't limited to integers
woven marlin
Take 2 as a factor,and we have 2(4A^4+A^2+-1/2)
Or 2((2A)^2-1/2)^2+1/2
amber veldt
or quadratic formula, heard of that?
amber schooner
i thought its different since the leading degree is 4
(x^2)^2 = x^4
it’s a quadratic in terms of sin^2 x
nimble blaze
if you want to consider that,
you'd want to test +-(factors of the constant)/(factors of the leading coefficient)
amber veldt
but theres no reason to
nimble blaze
which here would also include +-1/2, 1/4, 1/8
amber schooner
it’s pretty straightforward to factor
amber veldt
because you easily find two numbers that add to 2 and multiply to -8
silk crescent
or quadratic formula, heard of that?
got -1 +/- 3 divided by 8
silk crescent
because you easily find two numbers that add to 2 and multiply to -8
but i thought that only works if the leading degree is 2
amber veldt
let B=A²
silk crescent
if its over 2 its different
amber veldt
then this is 8B²+2B-1=0
nimble blaze
what you have is a disguised quadratic
to avoid doing two subs, subbing for sin^2(x) instead of sin(x) would be more efficient
silk crescent
ok did that
silk crescent
got the answer
nimble blaze
Why am I wrong?
wrong signs
silk crescent
didnt think i could substitue A=sin^2 so got confused
just subbed it as sinx
but yeah makes sense
woven marlin
wrong signs
Oh
silk crescent
.close
odd edgeBOT
odd edgeBOT
Available help channel!
cold bane
How would I do these 2 questions
odd edgeBOT
How would I do these 2 questions
cold bane
I have no idea where to begin
quasi sparrow
Do you know polar coordinates
cold bane
How would I do these 2 questions
@quasi sparrow for 5, do we just rearrange and get x = 2sin3t and y = -2cos3t
then test each option?
final ether
give me a condition on the radius to be tangent to the y axis
final ether
cold bane
ok
is the condition what the sols says
the radius must be same distance from center to the y axis?
cold bane
the radius must be same distance from center to the y axis?
@final ether
cold bane
How would I do these 2 questions
So both these questions just basically need to use pythagorean theorem
so we can put x^2 + y^2 = cos^2 x + sin ^2 x etc.
ight
.close
odd edgeBOT
odd edgeBOT
Available help channel!
karmic stirrup
How do you factor? Without guessing and checking or working backwords from solving it
narrow crypt
Pattern recognition
karmic stirrup
Wdym
karmic stirrup
But how tf am supposed to do the problems to get the patterns if its impossible to solve
copper quarry
Can you provide an example
cobalt coral
yes
karmic stirrup
cobalt coral
How this is impossible
karmic stirrup
im not saying its impossible
cobalt coral
Oh
karmic stirrup
what im asking is
cobalt coral
so u get 3
copper quarry
there r methods to factorizing quadratics sir you dont have to guess
cobalt coral
there r methods to factorizing quadratics sir you dont have to guess
oh
Just a sense
karmic stirrup
what are the methods
cobalt coral
-5(-1) ~ -5+(-1)
vivid wyvern
Factorize the denominator
karmic stirrup
Factorize the denominator
your a genious
narrow crypt
The unit
vivid wyvern
U will be able to cancel out (x-5) from numerator and denominator
narrow crypt
Is prolly using algebraic manipulation to solve limits gng
cobalt coral
If u can't solve a quadratic
Simply apply the quadratic formula
narrow crypt
😭
karmic stirrup
i understand the quadratic formula
cobalt coral
Yes
karmic stirrup
but that is basicaly solving then unsolving
cobalt coral
U don't understand
copper quarry
Js learn how to factor. It ain’t rocket science
Lil bro just use newton raphson /s
karmic stirrup
Js learn how to factor. It ain’t rocket science
thats what im asking
narrow crypt
You’ll get used to it
cobalt coral
Then no need factor
narrow crypt
Lil bro just use newton raphson /s
Nah use the bisection method 
/s
cobalt coral
Just use a calculator whatever
copper quarry
vivid wyvern
js find two numbers whose product gives u the constant and whi sum gives u the middle factor
copper quarry
bro just watch a vid
karmic stirrup
js find two numbers whose product gives u the constant and whi sum gives u the m...
how?
copper quarry
..
they do??
u cannot tell me every single video u watched did not explain how to factorise a quadratic
karmic stirrup
yea
vivid wyvern
And the middle term is -6x
karmic stirrup
yea ik its simple with integers but like you almost never seen just integers
cobalt coral
u cannot tell me every single video u watched did not explain how to factorise a...
It just a process of solving simultaneous equation
Blud
vivid wyvern
yea ik its simple with integers but like you almost never seen just integers
Give us as example
karmic stirrup
like 3.02x^2+1/2x-3
vivid wyvern
Dis u js make it up
cobalt coral
1/2x
karmic stirrup
all math is made up
karmic stirrup
no
narrow crypt
vivid wyvern
Sure in these case instead of digging a hole through ur head and wasting ur time
What u do is type that thing out in calc and find the values
And then use ur brain to figure the factors
cobalt coral
Sure in these case instead of digging a hole through ur head and wasting ur time...
Yes
narrow crypt
wtf are u guys doing..
cobalt coral
U got calculator
karmic stirrup
Sure in these case instead of digging a hole through ur head and wasting ur time...
i give up whats even the point of factoring if ur just gonna use the formula anyway
.close
odd edgeBOT
narrow crypt
The thing cancels nicely..
cobalt coral
i give up whats even the point of factoring if ur just gonna use the formula any...
U use factorisation method when u can factorised it out as integer
vivid wyvern
Exactly
cobalt coral
If u can't no need to bruteforce
narrow crypt
i give up whats even the point of factoring if ur just gonna use the formula any...
Denominator is (x-5)(x-1)
Numerator is 12(x-5)..
cobalt coral
Just use calculator no one force u to use factorisation
vivid wyvern
Denominator is (x-5)(x-1)
Numerator is 12(x-5)..
Yea
odd edgeBOT
Available help channel!
karmic stirrup
Just use calculator no one force u to use factorisation
my teachers do but ok also wdym calculator
odd edgeBOT
my teachers do but ok also wdym calculator
cobalt coral
my teachers do but ok also wdym calculator
Wdym
narrow crypt
There is no need for calculator
narrow crypt
Ffs
karmic stirrup
no
cobalt coral
Bro is jokin
karmic stirrup
let me clarify
copper quarry
,w factor 3.02x^2+ 1/2x -3
clever fjordBOT
karmic stirrup
dawg i cant use wolfram on a test
cobalt coral
Huh
wooden python
what was the original question
karmic stirrup
ok let me clarify the exact problem im having
narrow crypt
it’s just this @wooden python
cobalt coral
like 3.02x^2+1/2x-3
...
wooden python
how did you get 3.02 anything then?
cobalt coral
So it's the same anyways
If u can't factorise no one is forcing u just use formula
lmao
vivid wyvern
how did you get 3.02 anything then?
He js made an equation and asked how to get it easily if the equations were not integers
cobalt coral
Quadratic formula is applicable for any quadratic equation
wooden python
you first try direct substitution and get 0/0; this tells you that (x-5) is a factor of both num and denom
cobalt coral
Same goes as calculator
wooden python
FOR FREE!!!
you get one factor for free here
finding the other factor is not terribly hard from there
cobalt coral
finding the other factor is not terribly hard from there
Yeah
narrow crypt
like 3.02x^2+1/2x-3
Is this actually from the exam or u just made this shi up
vivid wyvern
Made
cobalt coral
Is this actually from the exam or u just made this shi up
Even the wrong ques anyways
narrow crypt
Then why are discussing if we know that it’s prolly useless?
karmic stirrup
i said it was made up like 40 minutes ago also give me a seccond to find what im looking for
narrow crypt
i said it was made up like 40 minutes ago also give me a seccond to find what im...
Yeah
cobalt coral
Bro can't create a question properly
narrow crypt
you first try direct substitution and get 0/0; this tells you that (x-5) is a fa...
Okay. As Ann told u, this is one way that u can use to establish when u should factorize in a limit
cobalt coral
I can create one too
wooden python
- if you DON'T get 0/0, then you don't even need factorization in the first place cause you already know the value of the limit (incl. NaN)
- if you do get 0/0, then you need to factorize but you have a leg up in it
karmic stirrup
ok for a situation like this were im trying to find the limit as x-> -2 how would i factor out a answer for this
cobalt coral
8455720x^2-6790x-35
cobalt coral
ok for a situation like this were im trying to find the limit as x-> -2 how woul...
Multiply
wooden python
ok for a situation like this were im trying to find the limit as x-> -2 how woul...
ok well step 0 is to plug in x=-2.
wooden python
what do you get when you do that?
karmic stirrup
0/0
wooden python
ok then you know that (x+2) is a factor of both top and bottom
num is 3(x+2)
denom is (x+2)(???)
karmic stirrup
ummm i would have no clue how to get that
cobalt coral
Just make it integer
cobalt coral
If u can't do factorisation directly
clever fjordBOT
cobalt coral
Simple as that
wooden python
k
wooden python
ummm i would have no clue how to get that
do you want the smart way or the dumb way
karmic stirrup
i want a way ill understand
cobalt coral
ok make the denominator be integer
wooden python
the dumb way is (x+2)(Ax+B) = 3x^2 + 5.5x - 1, expand and match coefficients.
cobalt coral
Multiply both numerator and denominator by 2
wooden python
the smart-ish way is to notice that A=3 by looking at the leading term, and B=-1/2 by looking at the constant term
wooden python
Multiply both numerator and denominator by 2
this can be done but is not strictly necessary here
cobalt coral
this can be done but is not strictly necessary here
Ur way is smart but not for him
He don't even understand how to factorise a simple quadratic equation like x^2-5x+6
karmic stirrup
well
wooden python
is this true
karmic stirrup
(x+2)(x-3)?
cobalt coral
Scroll the chat up
cobalt coral
(x+2)(x-3)?
See
karmic stirrup
no
wooden python
you need to review your factorization m8
karmic stirrup
i didnt read ur thing i thought it was the one a wrote eailer
copper quarry
you need to review your factorization m8
i told em to watch videos on it and they said they did but none of them "explained" how to do it
cobalt coral
When u can't even factorise it fast now u teach a more complicated way I'd say ()
karmic stirrup
(x-2)(x-3)?
i know how to factor simple things like that
the reason i got it wrong is becuase u copyed one a wrote eariler but u changed it but i thought it was the same
wooden python
have you heard of such things as:
- vieta's formulas
- polynomial division
- the concept of equating (or matching) coefficients
narrow crypt
karmic stirrup
ummm
cobalt coral
have you heard of such things as:
1. vieta's formulas
2. polynomial division
3. ...
Polynomial division not rlly used alot
karmic stirrup
the 3rd one sounds familuer
cobalt coral
Since u need to know one of the roots
wooden python
i was casting a wide net here.
wooden python
denom is (x+2)(???)
from here and below
i was trying to explain how to find the other factor
i.e. how to solve the following problem:
the polynomial 3x^2 + 5.5x - 1 has (x+2) as a factor. find the other factor.
i want a straight "yes, go over it again" or "no, shut your yap about it" answer
karmic stirrup
go over it again
wooden python
yes
the other factor is going to have degree 1
do you need that explained in detail or are we good
cobalt coral
u did like (x+2)(ax+b)
He guessed a root
karmic stirrup
i dont know what degree 1 means
wooden python
He guessed a root
who's "he"?
wooden python
i dont know what degree 1 means
do you know what "degree of a polynomial" means
cobalt coral
i dont know what degree 1 means
Power of 1
wooden python
He guessed a root
who's "he"?
cobalt coral
who's "he"?
U
wooden python
i'm a she.
cobalt coral
In order to do polynomial division u need to guess one root forst
wooden python
i'm a she.
wooden python
why tf are y'all skulling me over this.
cobalt coral
some kind of unexpected
wooden python
karmic stirrup
cuz he is a non gender specific term for the most part when typing
bitter folio
some kind of unexpected
you can see the pronouns
wooden python
HE IS NOT A NON-GENDER-SPECIFIC TERM
bitter folio
like i have a blue diamond and i am he/him
wooden python
IT IS CLEARLY FUCKING MASCULINE THANK YOU VERY MUCH!
bitter folio
pls dont misgender people in future thanks
karmic stirrup
if u dont know or think about someones gender everyone just says he or him
cobalt coral
pls dont misgender people in future thanks
What u supposed me to do
bitter folio
What u supposed me to do
check pronoun roles in the future
cobalt coral
Say 'it'?
karmic stirrup
it
wooden python
no
bitter folio
but lets get back to the question
wooden python
they/them
wooden python
is the default if you don't know someone
bitter folio
Or ask ur gender before chatting
you can already see it lmao
pink diamond or blue diamond
or any other colour
cobalt coral
They means everyone not include urself
karmic stirrup
look ive been banned from a discord before for calling someone a them
cobalt coral
pink diamond or blue diamond
Colour defined gender,
bitter folio
but lets get back to the question
.
cobalt coral
This is 2025
wooden python
pink diamond means she/her, blue diamond means he/him, yellow means they/them, grey means anything goes
cobalt coral
Not 1005
wooden python
ok should i call the mods
bitter folio
Colour defined gender,
its a self role
karmic stirrup
no plz
cobalt coral
Oh
wooden python
or can we all put our big girl/boy/etc. pants on and get back to the question?
karmic stirrup
i mean u were the one complaining but plz do go ahead im still confuzeld
wooden python
DON'T CALL ME BRO
cobalt coral
What
wooden python
<@&268886789983436800> this is like the 3rd offense now
karmic stirrup
bro is 100% gender neutrol
cobalt coral
I thought bro usable on every gender huh
wooden python
well newsflash no the fuck it don't.
bitter folio
ann dosent like it -> dont use it on her
karmic stirrup
ive called my mom bro like 50 times
bitter folio
simple
sinful grove
if u dont know or think about someones gender everyone just says he or him
This is very bad
cobalt coral
anyway my point is
i didn't **guess** a root.
Yes u did
wooden python
no
cobalt coral
Then
karmic stirrup
This is very bad
its not wrong most people do it but anyway enough of it
wooden python
no 6th sense no nothing
cobalt coral
Then how u got it
wooden python
i know it's a root because plugging x=-2 in yielded 0/0
and the REASON we bothered to do that is that we're finding the limit as x->-2
karmic stirrup
oooo u smart that actualy makes sense
cobalt coral
For some question it doesn't mean that x=-2 is a root
narrow crypt
This is very bad
generic he used to be taught
cobalt coral
It's not applicable for every question
wooden python
therefore it is not a guessing matter at all that -2 is a root, capisce?
cobalt coral
Yk
wooden python
It's not applicable for every question
for rational functions it is.
bitter folio
For some question it doesn't mean that x=-2 is a root
i don't think we understand your point
karmic stirrup
so if i have a limit for x->a can i factor out (x-a) comfortably?
viscid flint
(i think everything's already been said but yea pls respect people's choices & differences, especially here where we make it nice and easy with icons)
cobalt coral
U can direct know the root by substitute into the eq and get 0/0, bcz u are doing a school question
wooden python
so if i have a limit for x->a can i factor out (x-a) comfortably?
let me be more precise about the conditions
cobalt coral
So it's basically a pattern for question
karmic stirrup
umm
cobalt coral
Concept? No
wooden python
if:
- you are working with a limit as x->a, AND
- the numerator or denominator is a polynomial, AND
- you get 0/0 when plugging x=a in,
then yes, that polynomial can have (x-a) factored out of it.
karmic stirrup
the blue diamond assumed im male help
cobalt coral
Trick yes
karmic stirrup
if:
* you are working with a limit as x->a, **AND**
* the numerator or denominat...
makes sense
wooden python
the blue diamond assumed im male help
you have a blue diamond yourself. if you need to fix it, then you can go to channels & roles (at the top) and get yourself one that fits you better.
karmic stirrup
wait
viscid flint
sigh
karmic stirrup
for the num and denomiator is it either both or one has to be polynomial
wooden python
whichever one is a polynomial, that's the one you can factor (x-a) from.
it may be that you have a polynomial on the num and something else on the denom (or vice versa)
what im highlighting here is specifically the thing where like
the context of the limit question lets you know a root, and hence a factor, for free
WHY AM I STILL GETTING SKULLED.
cobalt coral
the context of the limit question _lets you know a root, and hence a factor, for...
I can simply make the ques be no solution
How u gonna do
karmic stirrup
oh could u help me with a situtaion where u wouldent get a factor for free
wooden python
... what.
cobalt coral
Oh every polynomial has a root
viscid flint
@cobalt coral you don't appear to be helping. Leave.
cobalt coral
y
wooden python
@karmic stirrup can you stop reacting with nerd emoji and others all over my messages?
it feels disrespectful.
karmic stirrup
ok
wooden python
i'm trying to help you and you fucking clown on me?
i'm not your mate to banter with, jsyk.
cobalt coral
yi
wooden python
ok right
karmic stirrup
i understand
wooden python
so this one's got roots.
and there's a different technique that you will wanna use to make this one happen.
sinful grove
Why are we still letting this happen, theyre being very misogynistic and one of them is still expecting help. Like what the hell
karmic stirrup
im supposed to multyply both sides by root(2x-5)-1
wooden python
sqrt(2x-5)**+**1
karmic stirrup
yea
wooden python
yes
the denominator is just bare x-3 so there's no factorization to be done there
do you understand what to do after multiplying by the conjugate?
wooden python
Why are we still letting this happen, theyre being very misogynistic and one of ...
OP at least seems to be a bit more well-behaved after the modping.
as for troller123 i just blocked her ass.
karmic stirrup
well u would get 2x-5-1/(x-3)(root(2x-5)+1) right?
cobalt coral
Why are we still letting this happen, theyre being very misogynistic and one of ...
Bro just arguing for 20 mins cuz of 'he'
wooden python
well u would get 2x-5-1/(x-3)(root(2x-5)+1) right?
brackets.
cobalt coral
Also her concept is kinda wrong I just explain
wooden python
Bro just arguing for 20 mins cuz of 'he'
<@&268886789983436800> continued misgendering. i was exceedingly clear in not wanting to be called "bro", and user troller123 is choosing to keep ignoring that.
wooden python
well u would get 2x-5-1/(x-3)(root(2x-5)+1) right?
yes, you get $\frac{2x-5-1}{(x-3)(\sqrt{2x-5}+1)}$
clever fjordBOT
Ann
wooden python
numerator is 2x-6, a.k.a. 2(x-3).
karmic stirrup
yea so u have 2 over the root and +1 but thats where i get stuck
wooden python
at this point you just plug x=3 in...
you've gotten rid of the sources of 0 in both num and denom
karmic stirrup
but its still 2/0 init?
wooden python
the whole purpose of this conjugate rigmarole is to get rid of unsafe factors and replace them with safe ones.
karmic stirrup
thanks i just relized why it was going so array thanks for the help
wooden python
god ngl troller had some nerve.
i hope she steps on a billion legos.
but ok whatever.
!done
odd edgeBOT
If you are done with this channel, please mark your problem as solved by typing .close
karmic stirrup
.close
leaden flicker
Hi there, I'm developing a game that is a clone of breakout. I thought it would be a relatively simple game to clone and so far it has been. The biggest issue I've been having is in doing a form of continuous collision detection for the ball. Please forgive me for my sloppy use of mathematical terms, I'm not super confident on the language and terminology in English.
I've made a little illustration first of all to show the situation and how far I've gotten so far; I included it.
My ball is represented as the circle at point C with a radius of r, it is moving towards point D which I've represented as a vector V. On the way there, it will intersect the boundary wall, represented as the line AB. The intersection of the line CD and the line AB is marked as point E. I'm good up to this point.
The problem begins here. Now that I've found point E I know I can find the normal of the line AB and multiply it by r to create the triangle in the second image.
My question now is, how do I find the hypoteneuse and the vertical line so that I can find point Q and point X?
Thanks!
gritty spire
you have to use a ball not a circle right
oh no, i confused the game lol
OHHHH atari breakout
gritty spire
most game engines have collision detect right
leaden flicker
they do but that is not an option for me here as I am not using a game engine, and I want to understand this mathematically.
gritty spire
so i was confused by the question sorry i cant help 😦
odd edgeBOT
@leaden flicker Has your question been resolved?
woeful vine
Hi there, I'm developing a game that is a clone of breakout. I thought it would ...
So you are looking for the length vertical side of the green triangle in the 2nd picture which is the distance between E and X, correct?
leaden flicker
Yes, I suppose that's the same thing as knowing where Q and X are.
woeful vine
we have a right triangle where we know one of the sides (r) and an angle (θ), does any trigonometric formula come to mind?
leaden flicker
Well I can vaguely remember that I had to solve this exact problem in high school but that's over 15 years ago. So no, they don't; it's all on the tip of my tongue.
woeful vine
in general it is true that
side = hypotenuse ⨯ sin(opposite_angle)
side = hypotenuse ⨯ cos(adjacent_angle)
and if we compare two sides
side = other_side ⨯ tan(opposite_angle)
leaden flicker
Hmm, so in this case hypotenuse = tan(θ) / r?
woeful vine
if you notice there's never hypotenuse and tan(angle) at the same time in any of which I wrote
leaden flicker
ah right
then it would be vertical = r * tan(θ) and then with thathypotenuse = sin(θ) / r?
woeful vine
you can use any formula to compute the one point you really want, but you have to use them correctly
in our case the side of length r is adjacent, and not opposite, to θ so the relation between r, θ and the hypotenuse goes through cosine, and not sine
r = hypo ⨯ cos(θ) --> hypo = r/cos(θ)
leaden flicker
ah..
woeful vine
but the other you wrote is correct
leaden flicker
Think I need to refresh myself a bit on the trigonometric functions, thank you :)
woeful vine
Think I need to refresh myself a bit on the trigonometric functions, thank you :...
(you can type .close to close the channel if you don't need any more help)
odd edgeBOT
odd edgeBOT
Available help channel!
mystic saffron
that is 5, please verify
odd edgeBOT
that is 5, please verify
twilit mulch
that is 5, please verify
Pythagorian
mystic saffron
ye
south plume
that is 5, please verify
what is the area of the triangle, in terms of r?
mystic saffron
r^2
twilit mulch
r^2 + 4r^2 = 25
twilit mulch
Nvm r^2 is the area
twilit mulch
Pythagorian too
twilit mulch
epic
brittle plinth
quick formula
twilit mulch
Use similiar triangles
mystic saffron
ok
brittle plinth
BX^2 = XA.XC
mystic saffron
i see 3
twilit mulch
BX^2 = XA.XC
mystic saffron
what is the dot in XA.XC
twilit mulch
It means multiplication
mystic saffron
oh
twilit mulch
Its popular in asia
twilit mulch
yea dot means multiplication
twilit mulch
Epic
mystic saffron
for this i have an idea
find the height
and then calcualte legs
so height is
18
twilit mulch
mhm
twilit mulch
How would you know the height intersects the mid point of given base
mystic saffron
twilit mulch
Ya
clever fjordBOT
✪Royal✪
mystic saffron
ye we need someone to verify
mystic nova
$6root(13) + 6root(13) + 24 = 12sqrt(13)+24$
"\sqrt{}"
mystic saffron
mb
brittle plinth
$6 \sqrt{13}+ 6 \sqrt{13} + 24 = 12 \sqrt{13}+24$
clever fjordBOT
C∆rbonizeD
mystic saffron
for this question i bet u use similar triangles
brittle plinth
yep
brittle plinth
actually there's a way that you don't need to use similar triangles
brittle plinth
yea sure
mystic saffron
brittle plinth
go on
brittle plinth
yep
brittle plinth
exactly
clever fjordBOT
✪Royal✪
