precalculus

[vestal zenith]

but did your teacher teach you how to make a regression?

[viscid thistle]

no

I have no idea what to do

[vestal zenith]

I can show you how to do it on wordmat

[viscid thistle]

ok

[vestal zenith]

so this is what I get

first you make a table like the one in the picture

and then you choose the wordmat tab > regression > exponential regression

and you get the following equation

[viscid thistle]

huh

Hm

Do you know how to do it on a graphing calculator?

I have to find the a and b values though

[vestal zenith]

yes you can see the a and b values in the equation

[viscid thistle]

I need a graph though

[obsidian monolith]

kyoto

[viscid thistle]

ok

[vestal zenith]

you can plug the equation in your graphing calculator

but I can see in your case the form is:

$y=a \cdot b^x$

[obsidian monolith]

kyoto

[vestal zenith]

so what's the value of a and b here? @viscid thistle

[viscid thistle]

I don't even know so confusing

[vestal zenith]

look at the picture I sent

[viscid thistle]

Can you show me it on paper @vestal zenith over text is hard

Ye

Idk how you got those numbers though

in the progress

[vestal zenith]

it's a bit hard to explain when you're not used to wordmat

and i'm wondering why your teacher haven't taught you how to do regression yet

i'm not familiar with desmos but I found a video about it here: https://www.youtube.com/watch?v=zcZaI-xfiFE

[viscid thistle]

ok ty

[thin jewel]

anyone help me with velocity problem?

[vague hearth]

Y = ab^x where a is the initial value and b is the growth factor

Or decay

[trail niche]

Depends

If x>1 it’s growth

If x<1 it’s decay

[rare zephyr]

is there such thing as the inverse of a factorial

like how division is the inverse of multiply

[prime yoke]

can someone simply explain the reasoning behind why floor(-x) and -floor(x) are different

[undone pawn]

n < x < n+1
=> floor(x) = n

-(n+1) < -x < -n
=> floor(-x) = -(n+1)

@prime yoke

[supple hazel]

yo im so lost right now

#33

[thorn agate]

Can anyone help me with a d solve it.

[late pewter]

which part are you stuck on

[mystic umbra]

@thorn agate

[thorn agate]

I dont know how to do it man.

If you could brief out the steps.

Will be helpful

[late pewter]

have you learned about derivatives?

if not have you learned anything about exponential growth

[thorn agate]

Yes

?

[late pewter]

rate of growth

thats just a derivative

sapphire grows overtime at a declining rate

thats also a derivative

[late pewter]

also this is a precalculus channel btw

[hexed granite]

if anyone knows any lmk

[hexed granite]

I need 7, or 8

[lunar axle]

can someone plz help

[proud raven]

do you have any notes?

@lunar axle

this is usually what youll use for C -> P

obviously youll need to simplify

[clever pecan]

r means radius

[viscid thistle]

yes.

[crude pilot]

yeah

[astral knoll]

yes, as it has changed cot into 1/cot which is wrong.

[viscid thistle]

Would the quadratic function be A(x) = 6^2 + 12x + c?

not even sure

@astral knoll

[astral knoll]

Area=width * length, so new area is (6+x)(12+2x), now multiply and u will get the result in the form asked.

[viscid thistle]

for a question like this

are we supposed to assume that the right triangles evenly share the height of the pole

so like each side thats adjacent to the pole is 15 ft

or am i getting that wrong

bruh

[viscid thistle]

@astral knoll Would it be (x+6) and (2x+12)

How you factor?

512=2x^2+24x+72

[crude pilot]

divide it by 2 then factor

[trail niche]

Probs not

Since they got different angles

Try treating the whole system as one big triangle by adding the two angles

You know the height of the base

And the angle of one side

Crap actually nvm then u can’t use trig

[worldly radish]

ping me 🙏

[steel tulip]

What have you tried so far?

[worldly radish]

well im just wondering what am i supposed to do with A, B, and C

[potent imp]

plug in -2 into the equation

[worldly radish]

-1 if x<0
-1 if x is greater/equal to 0

soo the 1st part is true but the 2nd part isnt

[potent imp]

yeah, because -2<0

so the first rule applies

so its 3(-2)+5=-1

[worldly radish]

and that would be the answer for a?

and youd do the same for the rest?

[potent imp]

yeah

[worldly radish]

what about this part here

[astral mantle]

what part of this are you confused about?

if x=3 then h(3) = 6

otherwise, just substitute x into the other equation

[potent imp]

also note that the other equation simplifies to x+3 if x is not 3

[quartz oxide]

help pls

[viscid thistle]

@quartz oxide do u still need help?

[quartz oxide]

yea

@viscid thistle

[viscid thistle]

ok, do you remember solving quadratic equations during Algebra II?

[quartz oxide]

yea kinda

[viscid thistle]

okay great, so simplify this let's just refer to this as $3x^2 - 7x + 2 = 0$

[obsidian monolith]

Jitter

[viscid thistle]

do you know how to put the equation into factored form?

[quartz oxide]

would it be (3x-1)(x-2)=0

[viscid thistle]

yep

so now we can put sin back in there, and we get $(3sin(x)-1)(sin(x)-2)=0$

[obsidian monolith]

Jitter

[viscid thistle]

we can set each equation equal to zero, getting the equations $3sin(x)-1=0$ and $sin(x)-2=0$

[obsidian monolith]

Jitter

[quartz oxide]

so is that the solutions for [0,2pi]?

[viscid thistle]

Well, you can simplify it and find an exact value by solving for $sin(x)$. first equation would be $sin(x)=1/3$ and the second $sin(x)=2$

[obsidian monolith]

Jitter

[quartz oxide]

ohh ok

thanks man

[thorn lily]

I need to solve for x (the original problem is circled). Below the problem is my work so far, but I’m stuck on how to get the x’s on one side of the equation. My teacher wants us to provide the exact solution and an approximate solution.

[shrewd root]

last step not so right

we have to "move" all the x to one side

[thorn lily]

Ok, what did I do incorrectly? And how do I move the x to one side?

[shrewd root]

u have to expand the brackets when it first appears

then move all x term to one side by addition/subtraction

then factor out the x

[thorn lily]

sorry, how do I expand the parentheses?

[shrewd root]

(x-1)log2: x log2-log2

[thorn lily]

Ok, I understand that part, thank you.

Is this right so far?

[shrewd root]

yes!

[thorn lily]

Thank you!

[shrewd root]

but not done yet

[thorn lily]

Yeah. Where do I go from here?

[shrewd root]

step 3^

[thorn lily]

By factor it out do you mean divide?

[shrewd root]

2x + 4, factor out 2 becomes 2(x+4)

[thorn lily]

Ah ok

[shrewd root]

factor means take out common factor

[thorn lily]

My answer is in the box - did I do it right?

Thank you so much for the help!

[shrewd root]

looks good

[astral knoll]

yes

[sinful gull]

What do you mean..

[onyx gazelle]

Never mind I figured out my own question

[sinful gull]

Alright then

[iron pendant]

https://notevensweating.shx.gg/4ZxiOV

can someone help me with this?

<@&286206848099549185>

[trail sun]

@iron pendant

[agile owl]

how would i answer this?

[sick steppe]

polynomial of degree n will have nth differences be constant @agile owl

[agile owl]

im not sure what that means

[sick steppe]

Lines (degree 1 polynomials) have constant 1st difference
quadratics (degree 2) have 2nd differences constant
etc

[agile owl]

oh ok then that's degree 4 cuz its x^4?

[sick steppe]

yes

[agile owl]

that makes sense thank u

the word differences messes me up

then this would be 3rd differences cuz its degree 3?

[sick steppe]

yes

[agile owl]

how would i find what number it is?

[sick steppe]

d = an! @agile owl

[agile owl]

d=4(3!) ???

[sick steppe]

yes

[agile owl]

oh

omg i learn this in class and these r so confusing

[elfin arrow]

Can anyone help me solve this question? "A balloon drops 2 balls with a time of 3s between the drops, after 10 seconds the first ball reaches the ground, how high is the balloon? (tag me if u can help me out)

[serene river]

anyone?

[ember socket]

Hello

I am Solving a exponential function. I’m at 4^2x+9=12

So after subtracting nine on both sides I end up with 4^2x=3

What would be the next step ?

[main narwhal]

you would take log, base 4, on both sides

[ember socket]

So log each side ?

[main narwhal]

yup

[ember socket]

Ok thanks a lot I was stumped on that and was unsure on what to do

Logs are so confusing

[main narwhal]

the rules may take some getting used to

but it is fine more exercises or examples will treat it as 2nd nature

[ember socket]

Indeed I’m still getting used to them after a week of practice

[main narwhal]

perfectly normal take some breaks in between overwork will prevent you to have the "aha" moment

[oak skiff]

hey can anyone help me? i just cant rember how to go about this

[main narwhal]

Do you know what is sec x equal to?

[cobalt swallow]

@oak skiff you need to use this arcsinu + arccosu = pi/2

[serene river]

yo

??

this should be zero triangles

right

[shrewd root]

infinite, question is not well defined

[viscid thistle]

hm

wouldnt that be ambiguous case

then you use law of sines to figure out the sides and angles*

you can determine whether it’s obtuse or acute right after

[drowsy karma]

How do I do this?

[viscid thistle]

@misty imp try symbolab

it’ll show you steps

[misty imp]

it says i have to pay for steps :c

[viscid thistle]

hm

[misty imp]

it gives answer but idk rly how to do it myself

[viscid thistle]

lets see, uk how they give you steps but its incomplete

you could open it on another tab and put that part where it was incomplete

then it shows you the steps on how to complete it

@misty imp

I had that trouble back then too

lol

[misty imp]

ooh okay ill try

thank u :0

[viscid thistle]

ye, plus no one’s rlly around here
kinda depressing

@misty imp did it work

[misty imp]

yeaa it kinda worked but it has fractions for exponents

my teacher said no fractions

[viscid thistle]

ic

[misty imp]

didnt rly pay attention in class so idrk how to fix it

eek

[drowsy karma]

<@&286206848099549185>

Pls help

[smoky pagoda]

x + 3y = 12 and 3xy is as big as it can be. First thing to note is both numbers have to be positive : if only one is negative so will the product, but if both are negative the sum cannot be 12. Next to try and find y you might want to consider all multiples of 3 smaller than 12 and compare against a necessary x value. Compare these products and take the maximum (this is assuming the question wants integer answers)

[drowsy karma]

Okay thanks

[scarlet oak]

@misty imp check out the laws of logarithms. For this problem in particular pay attention to the product rule and quotient rule

[vapid mica]

i got 9.13

[brazen tundra]

i found the angle measure using inverse tan but what do i do after to find the weight?

[shrewd root]

i think u have to resolve the force/tension in the diagonal cable to balance out the horizontal force and vertical mass

[brazen tundra]

how exactly do i do that?

<@&286206848099549185>

[agile owl]

how do i find the answer do i plug in the coordinates?

[vestal zenith]

you can plot the function in your graph calculator and check it

[agile owl]

ok

what does it mean by negative?

[viscid thistle]

Huh?

That's not even close to the method they probably want them to approach this

@agile owl consider how can you express that a number is negative first

[agile owl]

what? like -1?

is it like xER and yER?

[sick steppe]

f(x) < 0 means f is negative

[agile owl]

ok i get that

i think it's (-inf, 1) and (2,3)

[sick steppe]

yes

[viscid thistle]

I was trying you to go "to express a number to be negative, x<0". Then you'd go f(x)<0

[agile owl]

-infty?

[oak skiff]

anyone free to vc to help me with 2 problems?

[bitter sandal]

hey, how do i find the root of a non-factorable cubic equation?

the equation is: P(x) = 5x³+21x - 2

[uncut mulch]

that's not a cubic

its a quadratic in which case you could use the QF

[bitter sandal]

its 5x³ sorry

the whole equation is: 5x³ + 14x + 7x - 2

[uncut mulch]

the presence of 14x and 7x separated like that is very suss

are you sure its not supposed to be 14x**^2**?

because that would have nice roots

[bitter sandal]

nopee its 14x + 7x

[uncut mulch]

typo in the question?

,w solve 5x³ + 14x^2 + 7x - 2 = 0

[bitter sandal]

prolly, but its still solvable right?

[obsidian monolith]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by ramonov#0921

[uncut mulch]

,w solve 5x³ + 14x + 7x - 2 = 0

[obsidian monolith]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by ramonov#0921

[bitter sandal]

this is what i got when i assumed 14x as 14x²

[uncut mulch]

solveable yes, but very ugly

[bitter sandal]

i think i'd go with 14x²

[uncut mulch]

use cubic formula and/or depressed cubic

[bitter sandal]

yeah it looks ugly

[bitter sandal]

will cardano's method work?

[uncut mulch]

fancy name related to what i stated

[bitter sandal]

oh..

lmaoo

but thanks

:)

[pale python]

when finding non-permissible values of a rational expression, only the numbers on the denonminator is what you're working with, right?

Yes

[uncut mulch]

for rational expressions, the numerator would be defined everywhere so yes you only need to consider the denominator when determining non-permissible values

[pale python]

Alright, thanks!

[bitter sandal]

how do I find the polynomial equation using the quotient and remainder?

[uncut mulch]

its better if you post the whole question

[bitter sandal]

okay, uh, "Write a polynomial division that has a quotient of x+5 and a remainder of -2"

[uncut mulch]

consider the quotient-remainder form:
P(x) = (x-a)Q(x) + R(x)

[bitter sandal]

ahh that one! thank you

[scarlet oak]

Hey all, is there a reason to use cardans method to solve cubic equations? Im using the "a0/an" method that I've seen which is easier. If it's worth the time, I'll learn the cardan method

[cobalt swallow]

well use whatever can be used to solve, there is no set method. Studying Cardano's method is useful nonetheless, because sometimes the usual techniques don't work , it is a method to find the real roots of general cubic equation so sometimes does come handy, it is kinda like the quadratic formula method.

[scarlet oak]

Does cardano's method always work?

Also, do cubic equations come up in calc 1 or 2?

I was having trouble finding a video to explain cardano's method; most of them are in hindi, and I've yet to learn how to speak it

[cobalt swallow]

https://brilliant.org/wiki/cardano-method/

https://people.math.osu.edu/derdzinski.1/courses/4552/4552-cubic-quartic.pdf

https://nptel.ac.in/content/storage2/courses/111101001/downloads/Lecture23.pdf

I think this is enough, just write cardano's method , and a load of stuff is on internet in english

@scarlet oak

[scarlet oak]

Thanks! I'll definitely check it out. @cobalt swallow

[tribal vapor]

given a recursive sequence definition

lets say you want to find like the 4th term

is there a way to find just that one term, without having to start from the first term and progressing until you reach the 4th?

[fringe quartz]

ye there is a way

[tribal vapor]

how

for example

[fringe quartz]

well

should be obvious that

every term is you just adding +3

to the number

so you want 4th term. just multiply the 3 by 3

add 9 to 15

you get 24

[tribal vapor]

thanks

would this one be 19 ^ 3 * 4?

[fringe quartz]

which term you want

4?

[tribal vapor]

yeah

[fringe quartz]

yes

if you are not sure you can check 1 by 1

[tribal vapor]

yeah i just verified it with some python

do you know what the recursive definition for this sequence would be?

[gaunt mason]

Hmmm, there might be many.
If you assume the form of
a_n + A a_n+1 + B a_n-2 = 0

Inputting exact figures, give you 2 equation of 2 variables, A and B.

@tribal vapor

[tribal vapor]

Can you elaborate on that formula

What's it called, I've never seen it before

[gaunt mason]

It's a classical recursive formula with 3 terms

You can have more or less terms, but this is like the most common one.

You can have for example
a_n + A* a_n-1 = 0

[inner iron]

i need some help

with these radian trig equations

[inner iron]

Solve the trigonometric equation for all values −3π≤x≤3π. 2sin1/3x-sqroot3=0

[desert axle]

hi I'm in elementary school how do I start learning calculus now I want to make my paretns proud

[amber laurel]

any knows what this means?

http://prntscr.com/w6wkmb

[uncut mulch]

expressing something in base $\red{a}$ would mean to express it in the form ${\red{a}}^b$

[obsidian monolith]

ℝamonov

[amber laurel]

so for b for example its 2^64^2-x?

[past meadow]

no

[clever pecan]

@past meadow why so?

[past meadow]

Because 2^64^2-x doesnt equal 64^(2-x)

And its also probably not even what they meant to write

[clever pecan]

I'm assuming you let 64^(2 - x) = 2^a, and solve for a in terms of x

for problem (b)

same for problem (a)

but just 3^a

[clever pecan]

nobody has spoken in this channel since?

lol

[quick kelp]

I guess...

I'm stuck with a question

[quick kelp]

suppose the graph of y=ax^2+bx+c has its highest point at (1,7) and it's y-intercept is -3. Then what is a, b, and c?

[viscid thistle]

@quick kelp are you still stuck?

[quick kelp]

yes sirr

do you have to use a two point slope?

[viscid thistle]

Consider what happens at the y-intercept

ie what's the x value that characterises it

[quick kelp]

mhmm

isn't the y-intercept the same as the value of c?

[viscid thistle]

Ok yeah ig you could think it that way

If you are in the form y=ax²+bx+c yes, as when the function hits the y-intercept, x is 0, so you'd notice that y=a(0)²+b(0)+c simplifies to y=c

[quick kelp]

ohh

[viscid thistle]

Then what's our c?

[quick kelp]

the y-interceptt

[viscid thistle]

I mean what's the value of the constant c

[quick kelp]

y?

[viscid thistle]

You are given the y-intercept, aren't you?

[quick kelp]

yes

[viscid thistle]

If you know the y-intercept, and we just established that c represents the y-intercept, what's our c

[quick kelp]

mhmm

-3

[viscid thistle]

Good job

Now we have y=ax²+bx-3

Ok next piece of info

[quick kelp]

ohh!

[viscid thistle]

What can you do with the info: "highest point at (1,7)"

[quick kelp]

you can plug it in

?

[viscid thistle]

Yes

And you get an eqn

Our goal is to make a system of 2 eqn's to solve for a and b

What's the other equation you can have

[quick kelp]

hm..

[viscid thistle]

Think about highest point

[quick kelp]

(h,k)

[viscid thistle]

Think about what does that mean

[quick kelp]

vertex?

[viscid thistle]

Yeah

And what are the characteristics of a vertex

[quick kelp]

it has a vertex form?

is that the other equeation?

[viscid thistle]

Do you know how to get the vertex if you have for example y=x²+x+1

This is just an example

[quick kelp]

do you use an equation form?

[viscid thistle]

I want to see if you are able to do it numerically

[viscid thistle]

Wdym by eqn form

Uh i really have to go, i'll be back in some time

[quick kelp]

oh it's alright

You really helped

I can go n from what you said

[viscid thistle]

Ok gl

[quick kelp]

you made it much clearer

thank you!

[viscid thistle]

I'll be here after if you still don't get it

Yw!

[fiery wave]

@quick kelp if it can help, recall that the x coordinate of a vertex (in the form ax^2 + bx + c) is -b/2a

[quick kelp]

yes

hmm

so..

-7/2(1) if i am not wrong?

[fiery wave]

No, you don't have a and b yet

You have x = -b/2a
You are given x. What is it?

[quick kelp]

is x

1?

correct me if i am wrong:(

if i am right

would it be

1=-b/2a?

[viscid thistle]

Yes.

[quick kelp]

okay

so far

the only i got is for c which is -3

[quick kelp]

how would this vertex equation be relevant if i may ask>?

[fiery wave]

Because now you have 2 equations

The one you got by plugging (1,7) into the rule

And the one you just got

You can find a and b that way

[quick kelp]

hm...

let me think this through

i'm so sorry I don't know how to continue

so I got 2 equations which is 1=-b/2a and

y=ax^2+bx-3

[fiery wave]

You're close

Plug (1,7) into y=ax^2 + bx - 3

[quick kelp]

7=a(1)^2+b(1)-3?

[fiery wave]

Yes

So 7 = a + b - 3

You now have two equations involving a and b. Can you solve?

[quick kelp]

Ohhh

okok im solving it rn

what I did

I got the value of a from 7=a+b-3

which yield to a=-b+10

then plugged the the value of that a to 1=-b/2a

and got a b=20

and used that value of b again to 1=-b+10

so i got a=-10 b=10 c=-3

b=20*

[fiery wave]

Looks good @quick kelp

[quick kelp]

Oh my god

thank you so much

thank you to both of u! Really appreciate the help!

And the time you gave for a stranger :))

[iron pendant]

https://notevensweating.shx.gg/sbwmhP i got this problem wrong but i dont understand why, can someone explain and help me find what i did wrong?

[harsh smelt]

@iron pendant are you familiar with unit circle?

[uncut mulch]

describe what you did

[lunar axle]

can someone plz confirm if this is right for me??

[viscid thistle]

Is this a test?

@lunar axle

[lunar axle]

no it’s homework

@viscid thistle

[blissful ridge]

Looks correct

[lapis bone]

hi pls help

[fossil crown]

From

#❓how-to-get-help

Rule 4

If your question has not been answered for a minimum of 15 minutes, you may use the Helpers tag once. Please do not try to bump your question using this ping unnecessarily. Do not abuse this ping. Do not individually ping users with the Helpers tag without their express permission.

[bleak torrent]

I'm stuck at d

[stuck lark]

we can do some algebra then recognize a derivative

[bleak torrent]

I have no clue what that means

[stuck lark]

$\frac{1-2\cos(x)}{\pi-3x}=\frac23\frac{\cos(x)-\frac12}{x-\frac\pi3}$

[obsidian monolith]

RokabeJintarou

[bleak torrent]

Ohhhhh

Thxxx uu

I got it

[stuck lark]

you're welcome

[bleak torrent]

Idk if it's right or not

[cobalt swallow]

it is not that the answer is wrong but the way you did it, firstly you seem to have used the fact lim x-> 0 sinx/ x =1, nothing wrong with that , but your second step is wrong, in the way that it does not lead to third step the way you did it , to get to the third , you don't do multplication in sin, rather you do it outside sin, and that you will multiply by (m+n)(m-n)/4 , both above and below ( I repeat outside of sin) , then you will get the third step as you have

@bleak torrent

and after that you use lim x-> 0 sinx/ x =1 , so the third step expression will directly be as k^2 will cancelled when you use that fact , so -2 (m+n)(m-n) /4 =
-(m+n)(m-n)/2 the answer you got ,

[bleak torrent]

I see

Thank u

[prisma glacier]

Hey! I'd like to simplify sqrt(16+sqrt(31))-sqrt(16-sqrt(31)) into sqrt(2) but I don't know how to proceed! Does someone have an idea? Thanks!

[uncut mulch]

you could try assigning a variable to your original expression,
square it in an attempt to simplify some of the square roots and see what happens

[bold knoll]

hi hi. quick question.
i came across this identity in my book, is there a name for it?
$a_0=0, {\sum_{i=1}^{n}{(a_i -a_{i-1})}} = a_n$

[obsidian monolith]

Biggs

[viscid thistle]

telescopic sequence

in a sense that a lot of terms vanish

[odd shell]

Hi, I'm a little confused on how to do this type of question:

The point (1, -2) is on the graph of f(x). Describe the following transformations on f(x), and determine the resulting point:

g(x)=2f(x)+3

[sick steppe]

@odd shell do you have the transformations from f that make g?

[odd shell]

@sick steppe yeah that is why i am confused

this is the only information that is given to me

[sick steppe]

ok so what does the 2 do? and what does the +3 do?

[odd shell]

well so the 2 would be a vertical strech? and the three a vertical translation up?

[sick steppe]

yep

vertical stretch by a factor of 2, vertical translation 3 up

Now determining the resulting point means you want to know g(1)

but g(1) = 2f(1) + 3, but f(1)=-2

[odd shell]

so then would it be g(1)=2(-2)+3

[sick steppe]

yes

[odd shell]

g(1)=-1

like that?

ok ty

[sick steppe]

yes

[bold meadow]

0/0 so L'Hopital's rule

[uncut mulch]

consider 1 = sin(pi/2)
and the limit definition of the derivative

[fathom phoenix]

hello can anyone help

[bold meadow]

a. 1st derivative

[viscid thistle]

use derivativees @fathom phoenix

or

use $x = \frac{-b}{2a}$

[obsidian monolith]

Shen

[fathom phoenix]

ok thanks

[bold meadow]

ax^2+bx+c
c determines the height, changing this doesn't change the maximum height?

[viscid thistle]

it doesn't change where the maximum height is

$x = \frac{-b}{2a}$ is where the maximum is

[obsidian monolith]

Shen

[viscid thistle]

according to a quadratic in the form $ax^2 + bx + c$

[obsidian monolith]

Shen

[viscid thistle]

changing $c$ changes the max value, it just doesn't change where it is (along the x-axis)

[obsidian monolith]

Shen

[viscid thistle]

If you want to solve for maximum height, you need to do:

$a(\frac{-b}{2a})^2 + b\frac{-b}{2a} + c$

[obsidian monolith]

Shen

[viscid thistle]

where you plug in $\frac{-b}{2a}$ for $x$

[obsidian monolith]

Shen

[fathom phoenix]

oh wow lol i didnt even think of this

ill go back and check

thanks again

[blissful gale]

help

[sour plinth]

use the arithmetic series formula $\frac{n}{2}(2a+(n-1)d)$

[obsidian monolith]

187

[blissful gale]

how do i do the sum 165 part

[sour plinth]

let it equal to 165

then you know the values of a and d

a=5 and d=2

so solve for n

[blissful gale]

i put =165 at the end?

do the ns cancel out

[sour plinth]

no

solve for n

$\frac{n}{2}(2(5)+2(n-1))=165$

[obsidian monolith]

187

[blissful gale]

bro how

[sick steppe]

with math

it's 1 eqn 1 unknown

[bright grail]

im stuck on this question

i tried finding the inverse but i got stuck trying to isolate

[signal pivot]

So the input of ln can't be negetive or zero i.e it has to > 0

\begin{align*}
e^{x^2-x}> 1\
\therefore \hspace{6pt} x^2-x>0\
\end{align*}
So that gives either x=0 or x=1

[obsidian monolith]

S k

[bright grail]

how did you go from e^x^2-x > 0 to the second one

removing the e

and how did it go from > 1 to > 0 @signal pivot

i dont understand how to jump to that step

[sick steppe]

e^0=1 @bright grail

[woeful fable]

Is there a reason why during addition identities and subtractions for trig functions, when you break up the inner values, it has to be from the lower value to the greater value?

[sick steppe]

@woeful fable specific example if possible?

[woeful fable]

cos(2pi/12+3pi/12) != cos(3pi/12+2pi/12)

[sick steppe]

They equal the same thing

addition is commuative

[woeful fable]

I did the angle formula

and they don't equal the same unless I have an algebraic error

[sick steppe]

You likely applied the identity wrong then

cos(A+B) = cosAcosB-sinAsinB

[woeful fable]

I did cos(pi/4+pi/6)
Which sums just
cos(pi/4)cos(pi/6) - sin(pi/4)sin(pi/6)

[sick steppe]

yeah

[woeful fable]

That's equal to (sqrt2/2)(1/2) - (sqrt2/2)(sqrt3/2)

then that's sqrt2/4 - sqrt6/4)

which is sqrt 2 - sqrt 6 / 4

[sick steppe]

brackets

[woeful fable]

wdym

[sick steppe]

(sqrt2-sqrt6)/4

[woeful fable]

I just distributed and simplified

yes

yea

And that's not equal to what it's supposed to be

if I did cos(pi/6+pi/4)

The top one is my result

the correct answer is the bottom one

[sick steppe]

You either added or removed a negative somewhere then

[woeful fable]

cos(pi/4) = sqrt2/2
cos(pi/6) = 1/2
sin(pi/4) = sqrt2/2
sin(pi/6) = sqrt3/2

angle addition formula for cos(x+y) is cos(x)cos(y)-sin(x)sin(y)

idk where im screwing up

[sick steppe]

pi/6 trig values are backwards

,calc cos(pi/6)

[obsidian monolith]

Result:

0.86602540378444

[sick steppe]

,calc sin(pi/6)

[obsidian monolith]

Result:

0.5

[woeful fable]

oh

lmao

ty

[bleak torrent]

I'm struggling at E

[bold meadow]

0/0 so L'Hopital's

[mystic umbra]

@bold meadow I am trying to work this out on my own but I am not understanding the step by step for e

[bold meadow]

well after using L'Hopitals' you can see that it just becomes:
$\lim_{x \to a} \frac{\sec^2{x}}{-\sin{x}} = - \csc{a}. \sec^2{a}$

[obsidian monolith]

usernamephobic

[mystic umbra]

@bold meadow in my book it says f'/g'

so I need the derivitive of (tanx-tana) as step 1 to find f'

derivitive wrt x

but there is an a there

[blissful ridge]

Divide numerator and denominator by (x-a) and you'll notice that each term becomes the definition of derivative at x=a

[mystic umbra]

ok that matches the book

probably harder for me because I forgot the definition of a derivitive

isn't that just the definition of the limit of the tangent line?

slope of the tangent line

[blissful ridge]

$f'(a)=\lim_{x \to a} \frac{f(x)-f(a)}{x-a}$

[obsidian monolith]

My Army is Frozen

[mystic umbra]

ok thx

[autumn marsh]

Need help solving f(x) = 0

[uncut mulch]

the solutions aren't pretty

[autumn marsh]

Yeah I saw it on the graph but don’t know how to solve it

I mean the results

[uncut mulch]

W-Lambert or something

its not really a pre-calc question

[autumn marsh]

Ohhh

I don’t understand that method the W-Lambert

[cobalt swallow]

there is 2 solutions only , that can be seen for -2e^(-x) = x-4 by "hand " drawing the graph

but yes finding them won't be easy

using W lambert is a good way to solve it

[autumn marsh]

Can you show me how?

[cobalt swallow]

it gives solutions W( ...)+ .. something, they are expressed in terms of the Lambert's function , but we need numerical values I don't think you need it, considering you asked it in precalculus

I don't know how to do it properly as well

Only looked it up when Ramonov told about it, really interesting.

many find approximate solutions through iteration

maybe using Newton's method

[autumn marsh]

Hmm... I’ll try using Newton’s method

[mystic umbra]

blackpenredpen on youtube does the W-lambert functions method

[bright grail]

can someone explain this? how did they get this

[astral mantle]

do it 5head way by using lhopital

cos(0)/1 = 1

[viscid thistle]

geometrical inequality

not my question, but how did they get this

[viscid thistle]

they used the lengths of the line segments

[viscid thistle]

desmos proves this, but how would you get this without the use of a graphing calc?

[green kernel]

divide the red inequality by sin theta, then take reciprocals (which flips the inequality signs)

[viscid thistle]

why does taking reciprocals flip inequality signs?

oh nvm

ok tyty

[winged cloak]

Idk where to put this...

Do you know how to simplify this?

[viscid thistle]

Have you tried anything so far? @winged cloak

[winged cloak]

@viscid thistle Nvm i did it it was dumb 🤦‍♀️ i just couldn't see it

[worn violet]

Can someone help me with: Graph the function f(x) = cos (x) and g(x)= 1/2 cps (2x-pi) ?

[viscid thistle]

What do you know about the graph of y=cos(x)

[worn violet]

Suppose you need to find the height of a tall building. Standing 20 meters from the base of the building, you aim a laser pointer at the closest part of the top of the building. You measure that the laser pointer is 4◦ tilted from pointing straight up. The laser pointer is held 2 meters above the ground. How tall is the building?

[wispy aurora]

Draw a diagram

[worn violet]

and

[echo wagon]

Show your diagram

[wispy aurora]

^

[echo wagon]

If your diagram is right, it's just a trig question

[wispy aurora]

Yup

A calculator related one tbf

But trig nonetheless

[worn violet]

can u help me with the diagram im confused

[wispy aurora]

Sure

What have you gotten so far?

[worn violet]

not much im generally confused about this

[wispy aurora]

ok, my recommendation is imagining it in your head. Imagine a building, and you walk 20 meters away from it. Now draw one line as the building, and the other line as the path you took 20 meters away from it

[worn violet]

ok

now?

[wispy aurora]

Got it so far? Now I recommend drawing a line from the point where you currently are to a point 2 meters above it. That is to simulate the laser pointer

[worn violet]

ok

[wispy aurora]

Now draw a line connecting the laser pointer to the building

[echo wagon]

And show your drawing, even if it's not done and you need help to continue

[worn violet]

yeah im not getting it

i need to see it

nvm

[viscid thistle]

i think you were the one i was helping who i asked for the diagram but you didn't post it

[echo wagon]

Maybe their phone is too old to take pictures and they're embarrassed to tell us

Don't be, only the mods here are elitist. The rest of us are nice.

[worn violet]

no no

i solved it

lmao

[worn violet]

i have the iphone 11

I just need help with

Graph the function f (x) = cos (x) and g (x) = 1/2 cos (2x-pi)

[viscid thistle]

i also said "what do you know about y=cos(x)" previously answering this q and you ignored it or i skipped it

[worn violet]

i apologize i must have missed it

@viscid thistle im not sure about it

[echo wagon]

You've never heard of cos before and know nothing about it?

[worn violet]

im talking about the problem

i know cos

[echo wagon]

Do you know its graph?

[worn violet]

No I don’t know how to graph it

Would it be 2 different functions or one?

[echo wagon]

f(x) and g(x) are two different functions

[worn violet]

So therefore 2 different graphs?

[echo wagon]

You need to know how to graph cos(x), and then apply transformations to it to graph 1/2 cos(2x-π)

... Yes

[worn violet]

Ok so like this?

[echo wagon]

Yes, I don't have a lot of time to help. Hopefully someone else can

Glad you can send pics now, that will help

[worn violet]

Haha

I know you don’t have enough time to help but would that be correct?

[viscid thistle]

if that's supposed to be the graph of y=cos(x), then sure that's fine

though it seems you took a pic from desmos

[worn violet]

Yes that’s the easiest way

Now I just need to graph g(x)?

[echo wagon]

...

[viscid thistle]

know your transformations

as said

[echo wagon]

So you just wanted help on how to enter it into desmos?

[worn violet]

Yes and No

[viscid thistle]

huh

[worn violet]

Because I’m not sure if there’s supposed to be on one graph or two different graphs

So now I just to graph g(x) on a separate graph right?

[viscid thistle]

..yes

why are you asking me if you are typing letters into desmos correctly?

[wispy aurora]

OOOOF

[viscid thistle]

Me who still hasn’t figured out mobile desmos

[viscid thistle]

don't blame you - computer is better ngl

[proud sparrow]

@viscid thistle stop crossposting

[viscid thistle]

Suppose your making a bioweapon that kills 1/3 of the population.

But you & your secret organization of 9 people decide to immunise prior.

If people realize that 1/3 of the population didn't die they'll know you have the vaccine.

How do you change the kill rate to ensure the 1/3 of the population still die even after 10 have been immunised?

Supposing the population has 90 people, in normal circumstances 30 people would die.

But you want to change that so 30/81 people will die.

So the formula is like (population*ratio)/(population - vaccinated people)?

Does this check out?

[proud sparrow]

@viscid thistle not sure what you are asking... oh, you want to change it to kill 30/81 of the people? looks reasonable...

[viscid thistle]

yea so it doesnt look suspcious when 9 of the 90 people in the population are secretly immunised

[viscid thistle]

I know that logs aren't defined for negative numbers
but is there any other reason as well we can't do this:
ln(1)=0
ln(-1.-1)=0
ln(-1)+ln(-1)=0
2ln(-1)=0
ln(-1)=0

let ln(x)=c, x<0 and c belongs to R
e^c=x
which is not true for any c belonging to R

[trim hemlock]

well its not that it isnt defined

but it isnt defined in the real number set

but log of a negative number can also be defined by a complex number

[potent imp]

^

using e^ix=cosx+isinx

or more generally e^(a+bi)=(e^a)(cosb+isinb)

[novel cargo]

I'm not sure if this the place to ask this

is there a precedence rule for factorial and multiplication?

$3n!$ is $(3n)!$ or $3(n!)$?

[obsidian monolith]

az

[tawny nacelle]

i think its the latter one that is more common

factorial takes precedence over multiplication

[novel cargo]

thanks

[tawny nacelle]

np

[novel cargo]

checked in Mathematica

they give factorial precedence as you said

[tawny nacelle]

nice

[serene heath]

you thought they were lying? smh

[tawny nacelle]

[harsh smelt]

i mean ! is unary operator

[viscid thistle]

!n! what ! comes first?

[jagged glade]

The exclamation mark

[tawny nacelle]

if u do it on both sides, it cancels out

[viscid thistle]

smh !n is subfactorial

[blissful ridge]

Whoa never knew that

[maiden pelican]

How

How do you solve this system of equations

Parameters are p, q, r and unknowns are a, b, c

[smoky pagoda]

What's troubling you with this system?

It can be solved by any standard methods for solving linear equations

Gaussian elimination will work just fine here for example

[rain notch]

How would you go about solving #13, just for sine (without calculator).

[rain notch]

<@&286206848099549185> sorry it's already been some time

[bold meadow]

$\sin{-115}=\sin{-25}=-\sin(25)$

[obsidian monolith]

usernamephobic

[rain notch]

Thank you. However, I need to obtain a ratio as an answer, so how would that be done. My teacher had not gone over this type of scenario, where the reference angles are not common like 30, 45, 60, or 90 degrees.

[uncut mulch]

-sin(25°) is exactly what they want

[rain notch]

Alright

There is no way to go passed that I assume

[uncut mulch]

the less special angles are 15°, 36°, 72°,75°

[rain notch]

I see

[uncut mulch]

i mean if you read the question, they want
stuff in terms of trig functions of the reference angle

so even IF you reference angle would be 30°,
you'd give an answer like -sin(30°) instead of -1/2

[rain notch]

oh alright

thanks, I was completely overthinking it

[maiden pelican]

I can't do it for some reason..

[cobalt swallow]

@maiden pelican , just solve it the plain elimination way, Using equation (1) and (3), cancel out p, getting an equation in q and r, which you can solve using eqn(2). Doing by gaussian elimination is really quick

[maiden pelican]

Yeah.. I can't do this for some reason, something is going wrong and I don't understand what..

Oh I'll do that

[viscid thistle]

the problem does not ask for a ratio

[rain notch]

Yeah I was overthinking it, because I thought it correlated to earlier sets of directions from another worksheet

[viscid thistle]

Oh I see

[echo wagon]

@mighty talon Bruh, did you join just to spam your dumb competitions in every channel daily?

[viscid thistle]

Doing @ Moderators might be worthy here

[echo wagon]

Idk how they could have missed it

[peak badge]

I will do it for you.

<@&268886789983436800> - can we deal with an ad user here?

[odd shell]

https://cdn.discordapp.com/attachments/759248523479744523/793315052853264424/unknown.png

how would you determine the function given this table of values?

im stuck someone please help

[royal siren]

infinite

[sick steppe]

First thing you can do is figure out which of first,second,third, etc differences are constant

[median moss]

Would someone explain the concept of an infinite geometric series quickly please? Specifically its difference of finite geometric series?

[green kernel]

infinite geometric series go on forever, they have infinitely many terms

finite geometric series stop at some point, they have finitely many terms

does that answer your question? @median moss

[median moss]

I understand that part, I guess I should be more specific. I don't really get the divergence and convergence part. What role does the common ratio (r) play a part in the sum doesn't really click for me, even though its written there

[green kernel]

well, imagine trying to calculate that sum 1 + 2 + 4 + 8 + ...

by hand

it will just get bigger and bigger

[median moss]

yep

[green kernel]

it basically goes off to infinity

but when you have a sum like

1 + 1/2 + 1/4 + 1/8 + ...

since the terms are getting smaller and smaller

the more you calculate, the slower it will ‘grow’

so it will approach some specific value

[green kernel]

the terms r^a get smaller and smaller when you have the ratio between the terms is less than 1

[median moss]

Ahh I kinda get that, and it depends all depends on the common ratio I guess

[green kernel]

anything multiplied by a number less than 1 will be smaller

[median moss]

Give me like 10 minutes to just sit on it and try to actually understand and come back to you :)

Thanks so much

[green kernel]

anything multiplied by a number greater than one will be bigger

so in order for the terms to get smaller and smaller, you need to be multiplying by something less than one each time

ie r < 1

(at least for positives :p)

[median moss]

Ah got it, and because they get smaller term by term. The sum of these terms will just converge to a sum

[royal siren]

not because they get smaller term by term @median moss

it's due to the properties of geometric series - "how fast they get small"

for instance, 1 + 1/2 + 1/3 + ... gets smaller

but doesn't converge

[green kernel]

yeah, it’s quite subtle. for all geometric series, ‘they get smaller’ is a good enough intuition, but in general you can have a sum whos terms do not get small fast enough for it to converge

[median moss]

I'm probably using the terms converges and diverges loosely as its a calculus term and i haven't done limits yet. So sorry about that

[green kernel]

no dont worry, i just glossed over some details in my explanation

[odd shell]

@sick steppe yea the third differences are constant

https://cdn.discordapp.com/attachments/759248523479744523/793315052853264424/unknown.png

but what after that?

i know that one of the factors is (x-2)

but thats it

[sick steppe]

d=an!
constant difference = (leading co-efficient) (degree)!

so you know a(x-2)f(x)

[odd shell]

so a would be -1 right

what next

[sick steppe]

I guess just use the points and try and find f(x) @odd shell

[blissful niche]

Functions nice!

[delicate minnow]

Anybody here willing to teach me factoring tomorrow?

[proud sparrow]

what sort of factoring?

integer factoring or polynomial factoring?

[green kernel]

teach me how to factor integers in polynomial time

[proud sparrow]

I don't know, but I can teach you some O(n^(1/3)) and O(n^(1/4)) methods

[green kernel]

isnt that poly time

well, better than poly time

[proud sparrow]

n is the number to be factored

not the number of bits

[green kernel]

oh right

[delicate minnow]

Poly

[proud sparrow]

oh no polynomial factoring

https://en.wikipedia.org/wiki/Factorization_of_polynomials
this is crazy stuff

[proud sparrow]

do you have a class of polynomials you want to learn how to factor, or do you want to learn how to factor polynomials in general?

whoa LLL is crazy

[open vortex]

this might be something really obvious but how would I work out y - 1 as a function of x? Would this be written as f(x) = y - 1?
If it is, then I'm really lost. X is the inputs right, but the x's aren't being mapped to anything so how would there be a solution?

[echo wagon]

Do you have an exact question you are looking at?

[open vortex]

yep

The original question was find an expression for the function whose graph is the bottom half of the parabola x + (y -1)^2 = 0.
The hint was can you solve the equation for y - 1 as a function of x?

I'm a bit confused about the hint. (Also I wasn't tooo sure where to ask this as I'm starting calculus at uni but these were meant to be prerequisites)

[echo wagon]

@open vortex Oh, I see

Still here

,w plot x + (y-1)^2 = 0

[obsidian monolith]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by Lunasong#7197

[echo wagon]

Ping me if you come back

[open vortex]

@echo wagon i'm here

[echo wagon]

Okay, so you just have to isolate y in the equation

Make y the subject

So what do you think the first thing you have to do is?

[open vortex]

for the hint? How would I make y the subject if its currently f(x) = y -1, would that not be y = y -1

for the equation in the original question, i could rearrange and make y the subject

actually, i think i tried that but i got stuck at y^2 - 2y = -x -1 when I tried to make y the subject for the original equation

[echo wagon]

No

The equation is x + (y-1)^2 = 0

Isolate y in this equation

Oh, okay, you did do that

Didn't read all of it

What's the first step?

[open vortex]

ahhh i think i get it! So (y-1)^2 = -x , then square root the -x and add 1?

but can we square root the -x?

[echo wagon]

Yes

Why not?

x+(y-1)^2 = 0 doesn't have solutions where x is positive, so x <=0 and -x >= 0. So you are only taking the square root of nonnegative numbers

[open vortex]

what if x is positive, then we would be square rooting a negative number no?

ahhh

that makes so much sense!! Thank you so muchh!!

[echo wagon]

But you're not done yet

[open vortex]

im not?

[echo wagon]

What are the solutions to x^2 = 4?

[open vortex]

plus and minus 2

[echo wagon]

Right, so when you take the square root, ordinarily you should do +- in front

[open vortex]

ok so i have 1 +- sqrt(-x) ?

[echo wagon]

But y = +-sqrt(-x) + 1, isn't a function because it gives you TWO values of y for one value of x.

[open vortex]

ahhh i seee

[echo wagon]

Whereas a function should have one and only one value of y for each value of x

[open vortex]

so just -

[echo wagon]

Yeah

And if they said the top half of the graph, it would be +

[open vortex]

ahh, got it! Thank youuu :)))

[echo wagon]

Np

[astral reef]

U knw wut this server is friggin boring im leaving

[paper zenith]

ok bro have fun

[drowsy karma]

How did he get X + 2

I wasn't paying attention for one second

[astral mantle]

the pink goes with two consecutive even whole numbers

adding two to a whole even number gives a whole even number

[drowsy karma]

well so does 4

[viscid thistle]

if a function is a set of ordered pairs no two of which have the same first member, how are two to one functions functions?

wouldnt the first member be the domain?

[sick steppe]

x and x+4 wouldnt be consecutive if x is even

[drowsy karma]

am confusion @sick steppe

wym consecutive

[sick steppe]

consecutive means right after each other

[drowsy karma]

Ohhh

[sick steppe]

so 1 and 2 are consecutive numbers, but 1 and 3 arent

The question states consecutive even numbers

[drowsy karma]

i get it

so it has to be 2

[sick steppe]

so if we assume x is even, the next even number is x+2

since even numbers occur every other

you can also think of it as 2n and 2(n+1) for an integer n

[robust nest]

I need some help visualizing this problem and drawing it out

We just started these today and all I can do is draw the first part (12 nautical miles)

[robust nest]

anyone?

[sick steppe]

then you draw the 15 nautical line from where you ended

[mystic umbra]

this is vector addition

@robust nest 12 mi is the hypotenuse of a triangle

N60W is the angle of theta with respect to the y axis

,w 12e^(i*(5pi/6))

[obsidian monolith]

Results provided by WolframAlpha

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[mystic umbra]

see the decimal expansion is actually your x and y values

drop the i

if you do this twice you will have (x1,y1) and (x2,y2)

then if you do (x1+x2,y1+y2) which is the sum of the two vectors which is a vector that represents the relative location (final) compared to your starting location

[abstract garden]

hi can anyone tell me the steps for the answer?

[mystic umbra]

it has already been solved on a calculator

,w sqrt(0.688)

[obsidian monolith]

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[mystic umbra]

,w sin((236/360)*2pi)

[obsidian monolith]

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[mystic umbra]

notice that I use 236/360 to get the % of the rotation

,w 236/360

[obsidian monolith]

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[mystic umbra]

then we want to convert that % to radians

1 full rotation around a circle = 2pi radians

so I multiply by 2pi after I do /360 to get the conversion from degrees to radians

360 deg = 2pi rad

[abstract garden]

@mystic umbra how can we get that when we don't have calculator ?

[proud sparrow]

in general, you can't

unless you don't mind whacking by hand

[abstract garden]

I see

Thanks

[open vortex]

For this question, why is the answer 3? mostly the f(0) could take any value part

[echo wagon]

Well, if f(0) = k then because f is even we get f(-0) = k and there is no contradiction. But if g(0) = k then g(-0) = -k and this only works if k is 0

@open vortex

[open vortex]

Ahhh I get it now, thank you :))

[echo wagon]

Np

[viscid thistle]

if a function is a set of ordered pairs no two of which have the same first member, how are two to one functions functions?
wouldnt the first member be the domain? << don't mind the repost im just still confused on it

[echo wagon]

@viscid thistle A two to one function just means that there are two different first coordinates that give the same second coordinate

So the first coordinates are unique, the second coordinates are not

[proud sparrow]

{(0, 1) (1, 1)}

[viscid thistle]

ohhh for some reason I thought the second coordinates would be unique cause it's one i'm so dumb 😂

[proud sparrow]

that's when it's injective

[mystic umbra]

,w limit x->2 (x^3-8)/(x-2)

[obsidian monolith]

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[mystic umbra]

so this can only be done by factoring?

[bold meadow]

You can use Lhopital's too

[mystic umbra]

actually no

that is for the derivative right?

there are times when the limit exists but the derivitive does not exist

you can have a limit of 0

[astral mantle]

lhopitals works because you are dividing by 0

[mystic umbra]

but not a limit of l/0

oh ok

[astral mantle]

its 0/0

when you substitute

[mystic umbra]

is hopital's rule for limits or derivitives or both?

[astral mantle]

limits of indeterminate form

[echo wagon]

Lhopitals is to find limits, by taking derivatives.

[mystic umbra]

ok

[gloomy swallow]

i need help with dividing polynomials

is anyonee hereee

[harsh smelt]

[gloomy swallow]

ohh okay

so

it is

x-1/3x^4-6x^2+17x-8

[echo wagon]

What do you mean by dividing?

[uncut mulch]

$x-\frac13x^4-6x^2+17x-8$?

[obsidian monolith]

ℝamonov

[echo wagon]

Because the degree of the numerator is already smaller than that of the denominator

[gloomy swallow]

not like tha

wait a sec

[astral mantle]

$\frac{x-1}{3x^4-6x^2+17x-8}$

[obsidian monolith]

Star_

[astral mantle]

?

[gloomy swallow]

:\frac{3x^4-4x^3-6x^2+17x-8}{x-1}

$\frac{3x^4-4x^3-6x^2+17x-8}{x-1}

$\frac{3x^4-4x^3-6x^2+17x-8}{x-1}$