but the question says that after 7 years, the price has doubled
#help-26
opal vault
So would it be k/7
?
neon iron
so P(7) = ...
35k
opal vault
woah woah slow down
So, "after 7 years" -> meaning evaluating at t=7 (so P(7))
"The price has doubled" -> P(7) = Twice the original price
What is "twice the original price"?
neon iron
Oh ok makes sense
opal vault
Now we're left to find k...
opal vault
What do you think
yes it's ln(2)/7
neon iron
Thank you
opal vault
So P(t) = 5e^(ln(2)t/7)
topaz sinewBOT
@neon iron Has your question been resolved?
neon iron
yes it's ln(2)/7
For the second one do I just plug in 4 for t?
opal vault
For the second one do I just plug in 4 for t?
Yes sorry for late response
topaz sinewBOT
@neon iron Has your question been resolved?
brazen remnant
im trying to find a closed form of $$\sum_{n=0}^{\infty} \frac{f(n)}{p^n},$$ where $0<|p|<1$ and $f(n)$ is some polynomial, but im getting nasty stuff involving finite differences... does anyone have an insight on what i could use to evaluate this summation?
topaz sinewBOT
im trying to find a closed form of $$\sum_{n=0}^{\infty} \frac{f(n)}{p^n},$$ whe...
thorny flameBOT
lpieleanu
haughty mural
do we know anything about f?
brazen remnant
no
haughty mural
i will try to bruteforce it and maybe see something, let's see
brazen remnant
thorny flameBOT
lpieleanu
brazen remnant
then, i let $$S_i = \sum_{n=0}^{\infty} \frac{\Delta_if}{p^n},$$ where $$i \in {0,1,2,\ldots,\deg f}.$$ (we wish to find $S_0$)
thorny flameBOT
lpieleanu
brazen remnant
my only progress with this is that $$S_i(p-1)=p\Delta_if+S_{i+1} \forall i \in {0,1,2,\ldots \deg f - 1}.$$
thorny flameBOT
lpieleanu
brazen remnant
oh wait we also have $$S_{\deg f} = \frac{2ap}{p-1},$$ where $a$ is the leading coefficient of $f(n)$
thorny flameBOT
lpieleanu
topaz sinewBOT
@brazen remnant Has your question been resolved?
neon iron
topaz sinewBOT
neon iron
Doing part b and I need help pls
I just don't really know what to do
I guess u multiply 4 by ln2/7?
But idrk how
keen matrix
$4\frac{\ln2}{7}=\frac{4\ln2}{7}=\frac{\ln2^4}{7}=\frac{1}{7}(\ln2^4)$
thorny flameBOT
MrFancy
keen matrix
do you think you can figure it out from there? :)
topaz sinewBOT
@neon iron Has your question been resolved?
keen matrix
bro literally just solved it
keen matrix
What do I do next
well evaluate and interpret, 
but i'd start by simplfying more and putting it into a calculator cause you're not going to get a nice number
neon iron
Oh ok so basically just use a calculator got it
Ok cuz I wasn't sure if my professor would let me use a calculat0r
neon iron
I need help doing C
topaz sinewBOT
I need help doing C
neon iron
Please
topaz sinewBOT
@neon iron Has your question been resolved?
topaz sinewBOT
topaz sinewBOT
Available help channel!
mellow grotto
Hello, need some help with a some number theory and programming stuff. I am trying to solve this problem: https://projecteuler.net/problem=495
I have worked out so far that I can compute all the prime factors of N (adding 1 to the list), and then multiply them by each other to get K numbers out. So, in the example gives, 144 would have the factors of [1, 2, 2, 2, 2, 3, 3] and then we can multiply the numbers together to get one of the answers, eg 2*2*2 to get 8, and 3*3 to get 9, giving the list [1, 2,8,9] which is one of the answers. Is there an algorithm that can generate all of the lists? (either distinct or not, it doesn't matter)
topaz sinewBOT
Hello, need some help with a some number theory and programming stuff. I am tryi...
flat kindle
say N have m factors excluding 1 then we choose m-k numbers and multiply them
mellow grotto
Hm, I shall try it, but wouldn't that be too big? In the case of 100!, M = 239 and K = 10, then 239C10 is huge, one way is figure out how to deal with duplicates, since they are frequent, but I am not sure how to deal with them
flat kindle
wait lemme copy the question here
mellow grotto
One possible method would probably be to compute a list L that contains the unique factors, and another of the number of duplicates of each factors, and then compute the products somehow
flat kindle
let W(n,k) be the number of ways in which n can be written as the product of k distinct positive integers. (permutations of the integers are not considered distsinct.) Given W(100!,10) mod 1000000007 = 287549200, find W(10000!,30) mod 1000000007
computing W(10000!,30) seems impossible
mellow grotto
Hm possibly, but It has been solved before. I would be happy if I can do W(100!, 10), and let the code run for W(10000!, 30) or just thread it if possible.
For 100! there are only 25 unique factors (out of 239), and 1229 (out of 31985) for 10000!
Hm, if we only had 2 2s for example, we can either:
- Use 2 {And get 4} - This is most boring, but there isn't any other way to get 4, so the output from here is unique (Actually, you can derive this case from 2)
- Use 1 {And get 2} - But then, if you have Q solutions if you ignore the duplicate 2, do you have 2*Q - ??? {Some number} when you include the duplicate?
- Use Neither {And get None} - Now we have 2 2s, but we need to multiply them into the list, we can multiply any of the numbers by either 2 or 4, but what can we do with this?
I guess if you have some number of duplicates (D) of a factor (F), then you can use any power of F up to D, and then somehow distribute the rest into the list. There is then D - n duplicates to spread around
I will close this for now, but if someone is interested in helping please DM me!
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
true mural
topaz sinewBOT
true mural
how do i solve this? i dont understand the different variables
pls :P
can someone at least tell me what i need to look at to figure it out
topaz sinewBOT
@true mural Has your question been resolved?
odd flower
Are u sure that's written in english
topaz sinewBOT
@true mural Has your question been resolved?
stuck forum
still need help with this?
slate timber
please help with this question
topaz sinewBOT
please help with this question
slate timber
so initialy i tried doing log(a^log 7 to the base 3 ) +........+......
so that i would get loga(log 7 to the base 3 ) +.........................+................
can i use the identity
but i cant
<@&286206848099549185>
lapis raft
$(a^{\log_37})^{\log_37}=27^{\log_37}=(3^3)^{\log_37}=3^{\log_37^3}=7^3=343$
thorny flameBOT
TimK
lapis raft
$a^{log_ab}=b$
thorny flameBOT
TimK
slate timber
but c is 19 though
lapis raft
4 + 6 + 9 = 19
slate timber
how ?
lapis raft
wdym?
slate timber
343 + 121 + 5 = 469
how did this
slate timber
4 + 6 + 9 = 19
become this?
restive steeple
topaz sinewBOT
past cradle
made it 1 + 2tanx/(1-tanx)
Try inserting an e^ln in there
$\exp \lim_{x \to 0} \frac 1x \ln \tan \left ( \frac{\pi}{4} + x \right )$
thorny flameBOT
jewels!
topaz sinewBOT
To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.
pastel juniper
To ask for mathematics help on this server, please open your own help channel or...
.
topaz sinewBOT
@restive steeple Has your question been resolved?
topaz sinewBOT
topaz sinewBOT
Available help channel!
pastel nova
topaz sinewBOT
tacit nebula
can you send a more clear picture?
topaz sinewBOT
@pastel nova Has your question been resolved?
pastel nova
so
b1+b2=7/16 b1-b2+b3=7/8
find b1
and its a geometric progression
like
b2=b1*q
b3=b1*q^2
inner ravine
Find the value of x when the arithmetic mean of x+2 and 4x+5 is 3x+2. What formula should I use?
topaz sinewBOT
Find the value of x when the arithmetic mean of x+2 and 4x+5 is 3x+2. What formu...
frozen goblet
Maybe the arithmetic mean formula
pastel juniper
do you know what the arithmetic mean of A and B will be?
inner ravine
not yet
pastel juniper
<@&268886789983436800> GPT?
lone nest
Looks like it. And they have no non-GPT posts either.
pastel juniper
yeah
inner ravine
Find the value of x when the arithmetic mean of x+2 and 4x+5 is 3x+2. What formu...
my answer in this is 3, is it correct?
versed cairn
yeah
restive steeple
hard stuck on this, any idea how to start this?
topaz sinewBOT
hard stuck on this, any idea how to start this?
@restive steeple Has your question been resolved?
restive steeple
<@&286206848099549185> any idea
topaz sinewBOT
@restive steeple Has your question been resolved?
delicate burrow
what is the difference between
f = g
and
f(x) = g(x)
topaz sinewBOT
what is the difference between
f = g
and
f(x) = g(x)
radiant tapir
f = g , both are variables
I mean not necessarily. If f and g are functions it could just be a shorthand notation
neon iron
f(x) = g(x), are functions
delicate burrow
f = g , both are variables
no, not like variables. f and g are functions
neon iron
In crude language
neon iron
I mean not necessarily. If `f` and `g` are functions it could just be a shorthan...
Yeah unless specified, i should add
delicate burrow
neon iron
no, not like variables. f and g are functions
Then both of them mean the same, as mellowdramallama states
delicate burrow
in the 2.
it's asked if f = g. My doubt is does it ask if the domain is the same, or if the function when simplified is sthe same
because f(x) = g(x) when it is simplified.
But the domain is not the same
In some other exercices. it's asked if f(x) = g(x), so I am a bit confused
.close
fickle stirrup
In the right triangle ABC (with right angle at A), the inscribed circle touches the side AB at P and AC at Q. IF AP/PB = 1/2. How much is AQ/QC worth?
topaz sinewBOT
In the right triangle ABC (with right angle at A), the inscribed circle touches ...
What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
mint crescent
we have this so far right
I'm being lazy and saying AP=1
but recall that the triangle is a right triangle
So consider ||pythagorean theorem||
fickle stirrup
Ok I will try
Let's see, what I can take from here is that with these arbitrary measurements, QP = sqrt(2)
And lets call G the other point where the circle touches the triangle, I think that GP is 2, but I dont know how to prove it
still urchin
You can use that QC=GC too and apply pythagoras to ABC
fickle stirrup
But ok, How do you know QC = GC
still urchin
cuz they're both tangents same reason why AQ=AP
fickle stirrup
Yes but one is inclined it doesnt matter?
still urchin
can you show how you got there?
fickle stirrup
Yes first I expanded all
So, sqrt(9A^2P^2 + A^2P^2 + 2CAPQ + Q^2C^2) = QC + AP
So I did this, 9A^2P^2 + A^2P^2 + 2CAPQ + Q^2C^2 =(QC + AP)^2
Which is, 9A^2P^2 + A^2P^2 + 2CAPQ + Q^2C^2 = Q^2C^2 + 2CAPQ + A^2P^2
10A^2P^2 + 2CAPQ + Q^2C^2-Q^2C^2-2CAPQ-A^2P^2 = 0
Which gives, 9A^2P^2 = 0
But this is wrong of course
still urchin
You're dealing with AP as two variables a and p
It's just one variable AP you can let it be x if it confuses you while expanding
Same with QC
still urchin
Yes right
fickle stirrup
3x = y
still urchin
yea
still urchin
Yes right
fresh lava
is it possible to factor 2x^3-5?
topaz sinewBOT
is it possible to factor 2x^3-5?
fresh lava
because i'm trying to get the y-intercept of x^2-x-2 / 2x^3-5
topaz sinewBOT
@fresh lava Has your question been resolved?
odd jacinth
yes because it necessarily have a root (at least)
and at least one of them is easy to find
2x^3 - 5 = 0
x^3 = 5/2
one root is cubic root of 5/2
so it can be factored by x-cubic root of 5/2
(the other roots are complex)
but I don't see why you want to factor
topaz sinewBOT
@fresh lava Has your question been resolved?
icy pilot
I am confused and I hate it
topaz sinewBOT
I am confused and I hate it
icy pilot
Set the bottom to zero right
Finding domain
Then I square
Is that ok
Now I get 16 …
What happens with that minus
Because
You could think of it as -1\sqrt right
That - is part of the square root ?
Or is it just a minus ??
icy pilot
No
What if you don’t move the square root over to the right
Thats what am doing
U just root without moving it
still urchin
Negative will stay in one of the terms yes, you just expand normally like you'd do with (x-y)^2
still urchin
Oh no you can't do that it should be x^2-2xy+y^2 if you're expanding (x-y)^2
icy pilot
Isnt it u and not x y
still urchin
The -sqrt(u+7) is a term and the 4 is the other term so like the 4 is the x and the sqrt(u+7) is the y
icy pilot
This don’t make sense
still urchin
So 4^2-2x4sqrt(u+7)+(sqrt(u+7))^2
still urchin
No you can't just square each term individually
icy pilot
If u root sqrt(u+7) doesnt that take out the aqrt
still urchin
Yes but if the whole thing is under the sqrt
The 4 isn't so it won't work
That's why you should just subtract 4 from both sides before squaring
icy pilot
Fucking maths
still urchin
So when you square the sqrt(u+7) the root just cancels
icy pilot
So your saying rooting goes over everything
(4 -sqrt..)^2
So its better and easier I just move the 4 instead of having to foil
still urchin
Exactly
Before doing the ^2 tho
So 4-sqrt(u+7)=0
Move the 4
Then square both sides
still urchin
Ok if it was -sqrt(u+7) = -4
you could just divide by -1 so the negative cancels and it cancels when you square them anyways so it doesn't matter
icy pilot
How does it cancel when you square them
Is it basically two terms is that what you mean?
Or no not really
still urchin
How does it cancel when you square them
(-anything)^2 the negative cancels you're multiplying two negative numbers so the result is positive
still urchin
16u
icy pilot
So both terms get squared separately?
still urchin
Yes
icy pilot
The -4 and the sqr
icy pilot
So 4u squared is 16u^2
still urchin
Yes right
icy pilot
Bc u square both factors
still urchin
Yea
icy pilot
Is there a rule that allows us to think like that?
still urchin
If they're multiplied you square everything if they're added or subtracted you can't
icy pilot
Ok ty, I hope I don’t have any more holes in my knowledge 
still urchin
Oh no problem dw
icy pilot
.close
neon iron
How do u take the derivative of -3 root(3t)
topaz sinewBOT
How do u take the derivative of -3 root(3t)
neon iron
$-3root3t$
thorny flameBOT
Drey 🔍
neon iron
Oop
noble laurel
$-3\sqrt{3t}$
thorny flameBOT
Austin
neon iron
Yeh
noble laurel
The derivative with respect to t?
neon iron
Yes
noble laurel
well recall the rule about constant multiplication
the -3 is a constant being multiplied so we can just ignore it and leave it out front, right?
neon iron
Yes
noble laurel
$\sqrt{3t}=(3t)^{\frac{1}{2}}$
thorny flameBOT
Austin
noble laurel
now looking at this, you might be able to see the power rule should be applied here.
and don't forget to apply the chain rule
$\frac{1}{2}\cdot (3t)^{\frac{-1}{2}} \cdot (3t)'=?$
thorny flameBOT
Austin
noble laurel
the inner function is (3t)
neon iron
Oh
noble laurel
as demonstrated above
neon iron
How do u take the derivative of root-1/2
noble laurel
you don't have to, but it would just be the power rule again. You already took the derivative of the square root when you used the power rule. Now applying the chain rule all that is left is th derivative of the inner function, which is just 3t
$$-3\sqrt{3t}=-3(3t)^{\frac{1}{2}}$$
Let the outer function be $f(t)=\sqrt{t}$ and the inner function be $g(t)=3t$, then,
$$-3(3t)^{\frac{1}{2}}=-3f(g(t))$$
Now what does the chain rule tell you the derivative of $-3f(g(t))$ is? Well it is $-3f'(g(t))\cdot g'(t)$. So what is $f'(t)$ if $f(t)=\sqrt(t)$? And what is $g'(t)$ if $g(t)=3t$? Combine those to find your answer in the form given above.
thorny flameBOT
Austin
noble laurel
@neon iron
noble laurel
How do u know which is outer and inner function?
I mean one is literally inside of the other one
so
neon iron
True
noble laurel
but if you want a better explanation it is by order of operations. You do exponents first, so square root is the outer (first) function, and multiplication, the inside, comes next
noble laurel
your f'(g(t)) is not quite correct
neon iron
f’ of root t would be like 1
noble laurel
no
noble laurel
f'(t) is the derivative of sqrt(t)
which is 1/2 (t)^(-1/2)
then you evaluate this at g(t)
because we wanted f'(g(t))
so since g(t)=3t
you get
neon iron
So root 2/t
noble laurel
no
neon iron
Wait
I forgot root
So 3 root 2/root t
I probably messed up on the negative exponent cuz I just did reciprocal
Sorry did I make u rage quit
My bad
topaz sinewBOT
@neon iron Has your question been resolved?
neon iron
<@&286206848099549185>
topaz sinewBOT
@neon iron Has your question been resolved?
topaz sinewBOT
topaz sinewBOT
Available help channel!
fossil minnow
I have a problem with the the one marked as red
topaz sinewBOT
I have a problem with the the one marked as red
half edge
Hmm that’s odd
fossil minnow
so I'm not crazy
sweet shard
Answers right. Probably just formatting issue
inner crater
How to find the Y intercept?
topaz sinewBOT
How to find the Y intercept?
inner crater
neon iron
You need help?
inner crater
I found X idk how to find Y
inner crater
You need help?
Yes
keen matrix
How to find the Y intercept?
the y-intercept happens when x=0
inner crater
Nice handwriting too
Actually? I think my handwriting is so bad
neon iron
Ye its better than mine
inner crater
the y-intercept happens when x=0
So how do you solve this I'm still confused
keen venture
@fossil minnow
I think a few people got fooled. It's u²/5.
keen venture
So how do you solve this I'm still confused
Take your original function. Let x = 0.
inner crater
Yeah I put 0 where there is X's
fossil minnow
<@155365771453792256>
I think a few people got fooled. It's u²/5.
oh ok i'll try that
fossil minnow
<@155365771453792256>
I think a few people got fooled. It's u²/5.
thanks you're right
knotty tapir
topaz sinewBOT
knotty tapir
I need some help with integrating fractions
I know how to integrate just not fractions
Can someone walk me through this
supple mist
Split it
knotty tapir
what do you mean
still urchin
x^2+25 is in both the numerator and denominator so you can subtract powers
knotty tapir
x^2+25 is in both the numerator and denominator so you can subtract powers
ah ok
so then it would become
2x/sqrt(x^2+25)
?\
knotty tapir
Two integrals*
so like
i integrate numerator sperately
and then denominator
separately
supple mist
What?
knotty tapir
?
still urchin
2x/sqrt(x^2+25)
Yes right
supple mist
2x/sqrt(x^2+25)
What? Thats wrong isn’t it
knotty tapir
yea
supple mist
Oh I thought there was a minus 💀
keen venture
u-sub
u = x² + 25
knotty tapir
u-sub
u = x² + 25
yes but it is just the fractions that confuse me, i took a year break in between math classes so i frogot some things
iirc
integrating fractions
was not as simple as splitting them like u can with derivatives
still urchin
2x/sqrt(x^2+25)
You can use the power rule here write it as 2x(x^2+25)^-1/2 it's multiplied by its derivative so you can use the power rule
knotty tapir
knotty tapir
You will have
√u / u
Which is the same as
1 / √u
Oh ok
I guess it makes sense yeah
What do i do with 2x tho
Again my confusion comes from not that I dont know integration but when it comes to fractions specifically
I just remember vaguely you cant split it first (then quotient rule) like u can with derivatives
knotty tapir
You can use the power rule here write it as 2x(x^2+25)^-1/2 it's multiplied by i...
So when we are doing this
I thought this was not allowed
still urchin
∫(f(x))^n . f'(x) dx = (f(x))^n+1 / n+1
The integral is in that form already so you can use that rule
knotty tapir
what if it is not in that form?
still urchin
you try to put it in that form like we did with that question
knotty tapir
∫(f(x))^n . f'(x) dx = (f(x))^n+1 / n+1
ok and for this i am not sure i fully understand this form lol
if there is some function being multiplied by another function that is raised to some power n
then what?
its weird to understand for me from stuff like this lol
still urchin
A function raised to the power n multiplied by the derivative of that function its integral is equal to that function raised to the power n+1 divided by that new power (n+1)
knotty tapir
A function raised to the power n multiplied by the derivative of that function i...
oh ok
so then knowing this now
then -2(x^2+25)^-1/2
is the integral of the entire function
so for future reference then
knotty tapir
A function raised to the power n multiplied by the derivative of that function i...
the correct approach is to try and get it in this form
still urchin
the correct approach is to try and get it in this form
yes right
knotty tapir
wait wait
ohhhhhh
you rewrote
i see what u did
u rewrote it as -1/2 to get it to that form and ok
so then you get
2x(x^2+25)^-1/2
and then u integrate that?
still urchin
Yes right
topaz sinewBOT
knotty tapir
i see
ok i got answer right
but are u positive that i can always get it in this form
im just afraid that there might be some fraction i have to integrate that is not able to be written in that form
yk
still urchin
Not always some other problems may need to be done by parts or ln
topaz sinewBOT
Available help channel!
neon iron
topaz sinewBOT
neon iron
is this question mis-worded?
clear lodge
why would it be?
clear lodge
is this question mis-worded?
im having trouble thinking of a function that satisfies every single condition at the same time tho.
Are you asked for a function for each case, or one function for all at the same time?
worthy storm
what does "parent" function mean in this context
clear lodge
the simplest function of a type, i assume
for example, y=x would be the parent linear function, y=x^2 the parent quadratic, and so on
worthy storm
(c) and (d) do seem to contradict each other
clear lodge
nah, xsin(x) satisfies both b and c
worthy storm
c and d, not b and c
how can a function be even yet have different behavior as x->+infinity vs -infinity
that doesn't satisfy d
clear lodge
oh, cd
topaz sinewBOT
@neon iron Has your question been resolved?
topaz sinewBOT
topaz sinewBOT
Available help channel!
stuck hatch
so I am working on alot of problems and I thought i understood cdf and pdf until I realized there was this expression P (X < or >= x) to define my F(X).... I know this means I dont have a clear understanding of cdfs and pdfs but idk why im misunderstood, here is the problem that threw me off:
topaz sinewBOT
so I am working on alot of problems and I thought i understood cdf and pdf until...
stuck hatch
actually really confused by this, I thought when we are calculating the F(X) (cdf) we are looking at the sum of the areas under the pdf, so why cant I just calculate the CDF and use the values of x where F(X) = 1/2 and F(X) = 3/4 and subtract each from 1 to get the values??
topaz sinewBOT
@stuck hatch Has your question been resolved?
stuck hatch
no
topaz sinewBOT
@stuck hatch Has your question been resolved?
topaz sinewBOT
@stuck hatch Has your question been resolved?
carmine lagoon
Just trying to figure out what to do from here
topaz sinewBOT
Just trying to figure out what to do from here
carmine lagoon
Always forget whenever theres a value before x^2
topaz sinewBOT
@carmine lagoon Has your question been resolved?
neon iron
Just trying to figure out what to do from here
hi
neon iron
after factoring, just divide by the coeff of x^2
like x+3/2 and x-8/2
like that
if you want i can send you after doing it in paper
carmine lagoon
Im just confused on formatting that
lone nest
If you have a procedure that works when the leading coefficient is 1, you can rewrite 2x²-5x-12 as 2(x²-2.5x+6) and factor inside the brackets.
carmine lagoon
I saw the answer i just dont know how to get it now
neon iron
Im just confused on formatting that
wait a minute
neon iron
If you have a procedure that works when the leading coefficient is 1, you can re...
exactly
wooden swift
what do u wanna do tho? factorise?
carmine lagoon
Completely factor
wooden swift
umm my guy tropo is typin smtg let's see
lone nest
I don't understand what's happening at the right side of the paper, but (x-3)(x+8) is x²- 5x-24, which doesn't immediately seem to be related to 2x²-5x-12.
carmine lagoon
Yea i multiplied the x^2 value to -12
lone nest
Though, hmm, if we take (2x-3)(2x+8) instead, we get 4x²-10x-24 which is exactly 2 times 2x²-5x-12.
And 2x+8 is easily halved, so we can write (2x-3)(x+4) = 2x²-5x-12. Still with the caveat that I don't understand where (x-3)(x+8) came from in the first place ...
neon iron
Though, hmm, if we take (2x-3)(2x+8) instead, we get 4x²-10x-24 which is exactly...
got it
neon iron
Just trying to figure out what to do from here
just middle term split -5x to -8x+3x
carmine lagoon
And 2x+8 is easily halved, so we can write (2x-3)(x+4) = 2x²-5x-12. Still with t...
Well usually we do something where we multiply 2x^2 by -12 and then we get the two values we were looking for and then we're supposed to divide or something
neon iron
Just trying to figure out what to do from here
2x^2-8x+3x-12
2x(x-4)+3(x-4)
carmine lagoon
?
carmine lagoon
Yea i got it thank you
neon iron
Yea i got it thank you
no problem
clever citrus
topaz sinewBOT
clever citrus
Could I provide a rref matrix for this?
so for one with no solutions I could give a rref matrix where the last row has 1 in it
drifting swift
there's nothing that stipulates that your matrix be in rref
nor anything that forbids it
clever citrus
so I can?
drifting swift
yes of course, why couldn't you
clever citrus
because I thought that would have been too easy
like for no solution I could just say
RREF such as:
1 0 0 0 | 2
0 1 0 0 | 5
0 0 1 0 | 7
0 0 0 1| 3
0 0 0 1 | 0
so the system of equation would be
x_1 = 2
x_2 = 5
x_3 = 7
x_4 = 3
wait acc
what would x_4 be here
it will be a free variable nvm im dumb
so x_4 = t
is that correct
topaz sinewBOT
@clever citrus Has your question been resolved?
clever citrus
<@&286206848099549185>
fading flume
clever citrus
tysm
and for infintely many solutions:
1 0 0 3 | 2
0 1 0 7 | 5
0 0 1 5 | 7
0 0 0 2| 3
0 0 0 0 | 0
could I do something like this
since rank < number of variables
@fading flume
and for unique solution same matrix but with rank = number of variables
1 0 0 0 | 2
0 1 0 0 | 5
0 0 1 0 | 7
0 0 0 1| 3
0 0 0 0 | 0
neon iron
1 0 0 3 | 2
0 1 0 7 | 5
0 0 1 5 | 7
0 0 0 2| 3
0 0 0 0 | 0
hmm
this looks like it could be reduced too
1 0 0 3 | 2
0 1 0 7 | 5
0 0 1 5 | 7
0 0 0 1| 3/2
0 0 0 0 | 0
neon iron
since rank < number of variables
but yes infinite solutions when rank is less than number of variables
clever citrus
but yes infinite solutions when rank is less than number of variables
wb my matrix for unique solution
thats correct right
neon iron
wb my matrix for unique solution
ye for a full rank
neon iron
let me check the rest one sec
clever citrus
let me check the rest one sec
wait
ill send my whole answer
i made a more
simple rref
my bad
neon iron
ill send my whole answer
okay
clever citrus
okay
wait isnt my first example incorrect now that I think abt it
since it can still be reduced further
r4 - r3
but it will have no solutions since it will look like
1 0 0 0 | 1
0 1 0 0 | 2
0 0 1 0 | 3
0 0 0 1 | 4
0 0 0 0 | -4
right
eh acc i think thats correct since its the same still
neon iron
For Unique solutions you'll have a full rank matrix like this right
$$\begin{pmatrix}
1 & 0& 0 & 0 & *\
0 & 1 & 0 & 0 & * \
0 & 0 & 1 & 0 & *\
0 & 0 & 0 & 1 & *\
0 & 0 & 0 & 0 & 0\
\end{pmatrix} $$
clever citrus
its 5 rows doe
clever citrus
I dont understand
clever citrus
right
but wont it be
4 variables
5 rows
so another column
of 0s
and one leading 1
in 4th column 4th row
and another row at the end of 0s
like I did
neon iron
one sec
clever citrus
like
is mine not correct
that wont be unique doe
for unique
number of rank = no. variables
thorny flameBOT
sen
clever citrus
righttttt
neon iron
finally
clever citrus
lol
neon iron
my point is
clever citrus
right
neon iron
For no solutions youd have something like this
$$\begin{pmatrix}
1 & 0& 0 & 0 & *\
0 & 1 & 0 & 0 & * \
0 & 0 & 1 & 0 & *\
0 & 0 & 0 & 1 & *\
0 & 0 & 0 & 0 & *\
\end{pmatrix} $$
thorny flameBOT
sen
clever citrus
whatttttt
neon iron
or another
neon iron
look at the last row
neon iron
0 + 0 + 0 + 0 = 5
neon iron
0 + 0 + 0 + 0 = 5
this wouldnt make sense
clever citrus
r4 - r3
neon iron
so no sols
clever citrus
I should have reduced mine further my bad one sec
neon iron
1 0 0 0 | 1
0 1 0 0 | 2
0 0 1 0 | 3
0 0 0 1 | 4
0 0 0 0 | -4
other rows may look different it doesnt matter but the key thing is the last row
that 0 0 0 0 | -4
then for infinite solutions
neon iron
For infinite sos youd have something like this
$$\begin{pmatrix}
1 & 0& 0 & 0 & *\
0 & 1 & 0 & 0 & * \
0 & 0 & 1 & 0 & *\
0 & 0 & 0 & 0 & 0\
0 & 0 & 0 & 0 & 0\
\end{pmatrix} $$
thorny flameBOT
sen
neon iron
correct
neon iron
you dont have free variables
neon iron
im assuming your last colum is augmented
clever citrus
right so what im saying is
from the system of equations only
how would one construct such a rref
with last row = -4
neon iron
how would one construct such a rref
the question didnt ask for rref but it makes it easy to answer
clever citrus
ye the question only asked for system of eq
but thats
where im stuck
like the system of eq
I gave
does it rlly
represent my rref
for this
neon iron
with last row = -4
that'd be for when you contradictory system
neon iron
and no solution
wdym no solutions
clever citrus
.
neon iron
no sols
$$\begin{pmatrix}
1 & 0& 0 & 0 & *\
0 & 1 & 0 & 0 & * \
0 & 0 & 1 & 0 & *\
0 & 0 & 0 & 1 & *\
0 & 0 & 0 & 0 & -4\
\end{pmatrix} $$
thorny flameBOT
sen
neon iron
wth is back sub
when you substitute the x4 you got to x3 and so on until you get solutions for everything
neon iron
anyways for you to get this
your last row must have had
0 0 0 1 | 0 (there could've been more ways)
such that R4-> -R3
1-1=0
0-4=-4
do you see what i did there
clever citrus
anyways for you to get this
your last row must have had
0 0 0 1 | 0 (there co...
thats
what i had
earlier
look
neon iron
ah perff
neon iron
perf perf
neon iron
what your system is saying is that x4=0 and x4=4
clever citrus
ohhh
neon iron
and state that x_4 = t
no no your x_4 is not a free var
at that point ig
clever citrus
shit idk I just came up with the rref
neon iron
idt they care abt rref
x1 + ... =
x1 + .... would suffice yes
neon iron
shit idk I just came up with the rref
k now try coming up with equations that would look like this
clever citrus
is that hard?
neon iron
nop
clever citrus
do I just go backwards
clever citrus
like the step we usually follow for rref transformation but backwards
neon iron
as long as you understand infinite sols, unique sols etc
clever citrus
o ur referrin to x_1 x_2 x_3 being anything
neon iron
your question wants 5 equations righ
clever citrus
right
neon iron
when you ref your 5 equations become rows right
clever citrus
right
neon iron
so for infinite sols you want rows that are linearly depedent
clever citrus
oh we r talkin about this rn?
neon iron
4x_1 + 5x_2 + 5x_4 = 10
8x_1 + 10x_2 + 10x_4 = 20
my example
do you see that eq 1 and 2 are depedent
neon iron
if you did ref on this
youd get something like
* * * *
0 0 0 0
neon iron
linearly dependating
not all just the ones you want to have 0
clever citrus
uhhh
neon iron
i hope im not bad at explaining
neon iron
4x_1 + 5x_2 + 5x_4 = 10
8x_1 + 10x_2 + 10x_4 = 20
see for this?
clever citrus
since i wasnt 2 rows of 0
clever citrus
see for this?
yes that gives one row os 0s
neon iron
yes yes
neon iron
oh we r talkin about this rn?
so you can make row 4 and 5 linearly dependent
neon iron
oh we r talkin about this rn?
see this row 1,2,3 are linearly indepedent
clever citrus
oh so
x1 + 2x2 + x3 + x4 = 1
2x1 + 4x2 + 2x3 + 2x4 = 2
6x1 + 12x2 + 6x3 + 6x4 = 6
for first 3 rows
neon iron
are they linearly indepedent
clever citrus
row 1 * 2 for row 2 and row 2 * 3 for tow 3 so all of them r
neon iron
x1 + 2x2 + x3 + x4 = 1
2x1 + 4x2 + 2x3 + 2x4 = 2
6x1 + 12x2 + 6x3 + 6x4 = 6
what would your ref look like for this then
if all them are linearly depedent
how many 0 rows will we have
clever citrus
one?
neon iron
no
clever citrus
or would it be 2
neon iron
right
clever citrus
ah so its number of linear dependant rows - 1
i dont usually at it that way
clever citrus
but is my current solution
for inf solution correvt
in the ss i sent
because I just realized
im missing a equation
neon iron
**sen**
but first tell me what you think about row 1,2 and 3
For infinite sos youd have something like this
$$\begin{pmatrix}
1 & 0& 0 & 0 & *\
0 & 1 & 0 & 0 & * \
0 & 0 & 1 & 0 & *\
0 & 0 & 0 & 0 & 0\
0 & 0 & 0 & 0 & 0\
\end{pmatrix} $$
thorny flameBOT
sen
clever citrus
x1 + 2x2 + x3 + x4 = 1
x2 = 2
x3 = 3
2x1 + 4x2 + 2x3 + 2x4 = 2
6x1 + 12x2 + 6x3 + 6x4 = 6
how abt that
clever citrus
bad?
clever citrus
but are row 1 2 and 3 depedent
what does depo mean
neon iron
it is
clever citrus
so its good?
clever citrus
and no solutions
uhhhh
idrk how to approach the no solution one
could I just put
5 equations that arent
dependant
at all
neon iron
idrk how to approach the no solution one
remember how we got no solutions
we had a contradictory system
neon iron
x_ = 4, x_4=something else
clever citrus
right so can I just not do these 5 equations then
x1 = 1
x2 = 2
x3 = 3
x4 = 4
x4 = 0
neon iron
should b false
give me an example of an eqauation like that
clever citrus
give me an example of an eqauation like that
0 + 0 + 0 + 0 = 4
neon iron
x1 = 1
x2 = 2
x3 = 3
x4 = 4
x4 = 0
hmm this is lazy do it the x_1 + x_2 way
clever citrus
hmm this is lazy do it the x_1 + x_2 way
right but how would I do that im so confused
like can i j put 4 non dependant equations
neon iron
2x_1 + 3x_2 = 0
2x_1 + 3x_2 = 16
clever citrus
and then one false at the end
neon iron
2x_1 + 3x_2 = 0
2x_1 + 3x_2 = 16
look at this
clever citrus
oh so how abt
this
alr then peep this
5x1 + 3x2 + 4x3 + 7x4 = 3
x1 + x2 = 3
x3 + 4x4 = 7
3x1 + 2x2 = 16
3x1 + 2x2 = 0
clever citrus
but how do i find
a ewuation
that would result in mine
my rrwf*
or would that be harder
neon iron
the rref you gave?
clever citrus
yes
neon iron
5x1 + 3x2 + 4x3 + 7x4 = 3
x1 + x2 = 3
x3 + 4x4 = 7
3x1 + 2x2 = 16
3x1 + 2x2 = 0
u just did
clever citrus
thats gonna result in that same one?
neon iron
5x1 + 3x2 + 4x3 + 7x4 = 3
x1 + x2 = 3
x3 + 4x4 = 7
3x1 + 2x2 = 16
3x1 + 2x2 = 0
row 4 and 5 are contradictory if reduced you'd get something like
0 0 0 0 | *
neon iron
thats gonna result in that same one?
we're creating new ones giving equations like your question asked
neon iron
5x1 + 3x2 + 4x3 + 7x4 = 3
x1 + x2 = 3
x3 + 4x4 = 7
3x1 + 2x2 = 16
3x1 + 2x2 = 0
you can ref this and show
clever citrus
ah
so when i rreg that
I get
1 0 0 0 | 0
0 1 0 0 | 0
0 0 1 0 | 0
0 0 0 1 | 0
0 0 0 0 | 1
neon iron
**sen**
it does not always have to follow this exact structure the last row 0 = something (non zero number) would suffice
neon iron
neon iron
2 dependant
uh
clever citrus
no bcz look
neon iron
what happens again when we have depedent
neon iron
right
clever citrus
what happens again when we have depedent
one of them turn 0
neon iron
one of them turn 0
perf
neon iron
so we will have 4 ranks
and we dont want contradictory
clever citrus
for sure
neon iron
so yes
neon iron
np np
clever citrus
.close
topaz sinewBOT