help-27

[reef basalt]

x-x?

[safe jasper]

you subtract the exponents when dividing only when the base is the same

so if i had a^m / a^n, that's the same as a^(m - n)

but in this case you have different bases

there's a different property that's useful here when the bases have the same exponent

if you have a^n / b^n, that's the same as (a / b)^n

[reef basalt]

Oh yeah

[safe jasper]

so based off that, what would this become?

[reef basalt]

(0.2/0.1)^x

[safe jasper]

and then you can easily simplify the inside

[viscid ivy]

El primoo

[reef basalt]

Oh got it

Thanks for the help

[safe jasper]

for pain and for glory!

[viscid ivy]

You play ?

[safe jasper]

not as much anymore but i used to play all the time

especially around 5 years ago near global release

i used to be top 3 leon mains in the country before they inflated the trophy system

[viscid ivy]

I used to i redownloaded

So many brawlers man !

[safe jasper]

fr i think it's like 80 now

[viscid ivy]

yeah

sometimes i get confused by skins

Which brawler !

[viscid ivy]

I had every brawler r 25 back then

[devout snow]

@reef basalt Has your question been resolved?

[devout snow]

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[haughty glen]

hi

[haughty glen]

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[tall stirrup]

steven0925 supremacy 🗿

[ancient rose]

bro figured it on his way here

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[scenic surge]

I dont understand the last line of this solution. I got everything else correct

[scenic surge]

what do they mean by degenerate at Beta

[devout snow]

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[scenic surge]

<@&286206848099549185>

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[steel sun]

can someone verify (9,2) being the answers for y?

[feral agate]

what are the corresponding solutions for x?

you have to state your solutions as ordered pairs of the form (x,y), just stating your answers for y is not sufficient

[steel sun]

oh

[viscid ivy]

,w solve x²-6y=7 and y²-10x+41=0

[woven radish]

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[viscid ivy]

@steel sun You can verify yourself this way

[feral agate]

huh, how would you even go about solving this?

[viscid ivy]

y = x²-7/6

[steel sun]

i took x as whole root 7+6y

[viscid ivy]

So with substitution

[feral agate]

that doesn't seem right.

[feral agate]

that would give a quartic no?

[steel sun]

most likely

[viscid ivy]

@feral agate as it has a real root

We will get an irrudicible quadratic and a linear factor

[feral agate]

what?

[steel sun]

cant u like

for eq 1

[feral agate]

it would be a 4th degree equation

[viscid ivy]

No bro

[steel sun]

take -6y from lhs to rhs

[steel sun]

and then also send away root to rhs?

[viscid ivy]

this says three roots so must not be a quartic

[feral agate]

forget about Wolframalpha for a second

[feral agate]

no

[lost laurel]

maybe by gaphing it

[feral agate]

$x^2=6y+7\implies x=\pm\sqrt{6y+7}$

[woven radish]

kheerii

[steel sun]

thats not allowed?

[feral agate]

you can't just take one of the cases

[steel sun]

thats not true?

[feral agate]

a more legitimate method is taking $$y=\frac{x^2-7}{6}$$ and substituting it into the second equation

[woven radish]

kheerii

[feral agate]

are you saying what I wrote isn't true

?

[steel sun]

no im just asking

is that supposed to be true or not?

[lost laurel]

wouldn't that give a 4th degree eqn though

[feral agate]

it definitely is

[feral agate]

yeah, it will

[viscid ivy]

Yeah its giving

[steel sun]

so why cant i use it?

[feral agate]

how would you deal with the plus-minus?

[steel sun]

wait why is there the plus minus

is that necessary?

[feral agate]

to make any kind of conclusion from that you need the plus-minus

$x^2=a^2\implies x=\pm a$, not just $x=a$

[woven radish]

kheerii

[feral agate]

so

that isn't a good method for this question

[steel sun]

oh

[feral agate]

or go the other way

[tall stirrup]

we can consider cases

[feral agate]

from the second equation get x as a function of y and substitute it into the first one

[steel sun]

ill have to sleep now cuz its 12 in my country and i have to wake up at 4

[feral agate]

either way you will get a 4th degree polynomial which you will have to deal with in some way

[steel sun]

.close

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[viscid ivy]

@feral agate it is a depressed quartic so maybe some method like cardano's or some formula

[feral agate]

depends on the quartic

we can safely assume that the resulting quartic would have some means of solving, since it's being asked in the question

,w simplify ((x^2-7)/6)^2-10x+41

[woven radish]

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[feral agate]

ah, there it is

@viscid ivy

[viscid ivy]

Yeah i see

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[lost crag]

is "sufficent and neccsary condition" meaning $A /leftrightarrow B$?

[woven radish]

Ayanokoji

[lusty sapphire]

yes

[lost crag]

ty mate

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[lost crag]

how do I prove a if only if ( ⇔ ) equality? for more context:
A⊆B<-->C⊆D

I give the actual thing I'm trying to prove too, but I mean what's the strategy to prove it

as in I know for A=B you have to prove A⊆B and B⊆A

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[small urchin]

$y=\sqrt[5]{(3x^2+2x)^2}$

[woven radish]

Penaldo

[small urchin]

I need to do the derivate and then simplify the result

[surreal night]

!status

[devout snow]

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[small urchin]

1. I don't know where to begin.

[surreal night]

do you know the chain rule?

[small urchin]

No

[surreal night]

then this problem is going to be quite hard

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[devout snow]

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[carmine orbit]

how do i solve this

[devout snow]

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[surreal night]

chain rule

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@carmine orbit Has your question been resolved?

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[dreamy palm]

Find the equation of the normal to the curve 𝑥𝑦2+ 𝑥2y = 4xy at the point (1, 3)

[weak cove]

[dreamy palm]

?

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[quaint citrus]

[devout snow]

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[deep vortex]

#proofs-and-logic message

[clever whale]

hi guys anyone knows how to do this

[restive river]

6th or 7th

I mean i or ii

Ok so get the center first

Take reflection about y=1
You know how to do this?

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[mental crag]

1+1

[dusky drift]

No trolling pls

[mental crag]

ok fine

[dusky drift]

.close

type it

[quaint citrus]

bro does not have the power vested in him

[fierce heath]

bro got all the pronouns

incredible

[dusky drift]

try isolating the variable by transposing 5 to rhs.

then transpose 3

solve it

u get the value of p

[mental crag]

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[tacit cloud]

How can i prove that the triangle FXE is an isosceles triangle

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[glacial helm]

Hello

[glacial helm]

Can smn help with this?

[mighty galleon]

!status

[devout snow]

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[last prawn]

I have no idea to do this one Please help me to do it

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[subtle kindle]

can I solve this limit like this?

[feral agate]

no, not at all

[subtle kindle]

oh okay lol

[feral agate]

convert it into polar form

[safe jasper]

does taking logs help here?

it's been a while since i did multi

[tidal pumice]

2x+7

[safe jasper]

x^2y^2 = r^2?

[feral agate]

yeah

$x=r\cos\theta$ and $y=r\sin\theta$

[woven radish]

kheerii

[safe jasper]

i thought r^2 was x^2 + y^2

so how would it be x^2y^2

[feral agate]

ahhh wait

I messed up

I read it as x^2+y^2

[feral agate]

this probably still works though

[feral agate]

yeah you will have to

[subtle kindle]

you mean writing it as e^(1/(x^2+y^2) * ln(1+x^2y^2)?

[feral agate]

uhhh, yeah

the log part also in the power

[subtle kindle]

like that?

[feral agate]

e^(lim....), yes

[autumn path]

It's 0/0 form so u can use l'hopital's rule

[subtle kindle]

oh I see

because now we have a function of 1 variable

wait, but I did something wrong

shouldn't it be like this?

[safe jasper]

i don't think you can just evaluate the limits one at a time like that

[subtle kindle]

Isn't it how iterated limits are calculated?

[safe jasper]

this is basically just evaluating your function along particular lines

this approach doesn't always work

[subtle kindle]

I see

and when do I know that it works? Or I should use it?

[devout snow]

@subtle kindle Has your question been resolved?

[subtle kindle]

wait, isn't it already solved? the limit of the function f(x,y) is 0, but we have e^lim(x,y)->(0,0)...., thus it's e^0 = 1?

[feral agate]

it isn't an iterated limit, it's a two-variable limit

there's a difference

[subtle kindle]

but my teacher used it here

isn't it the same thing?

[feral agate]

what the teacher is showing you here is that, along two different paths the limit is coming out to be different

hence the limit doesn't exist

from this method you can prove a limit does not exist but you can't evaluate a limit like this

[subtle kindle]

oh I understand

sorry

Is this it?

.close

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[tender lotus]

$y''+k^2y=0\ y=Ae^{ikx}+Be^{-ikx} \text{ OR } y=Ce^{ikx}$ ??

[woven radish]

Adam Ch.

[pastel pasture]

isn't that just a simple harmonic oscillator equation

[tender lotus]

yes but which solution is correct ?

[pastel pasture]

well

technically your second one is just your first one with B = 0

is there a need to write the solution as a complex exponential

I mean I guess technically both would be write if you play around with the constants

[tender lotus]

yes

[tender lotus]

but the second solution doesn't imply the first

so the 2nd solution could be wrong to write it

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[pearl gorge]

What is the largest possible value for the sum of the squares of the sides of a triangle inscribed in a circle with unit radius?
I can't develop after finding using sine law:
a^2+b^2+c^2 = 4(sin^2 (A)+sin^2 (B) + sin^2 (C))

[lunar harbor]

consider using AM-GM

[pearl gorge]

I didn't achieve anything with this, could you show me how? I know there is a solution using the Lagrange multiplier

[lunar harbor]

$\frac{\sin^2 A +\sin^2 B +\sin^2 C}{3} \geq \sqrt[3]{\sin^2 A \sin^2 B \sin^2 C}$

[woven radish]

Civil Service Pigeon

[lunar harbor]

now recall when equality is achieved for AM-GM

[pearl gorge]

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[broken verge]

hi how do i find the range of arg z?

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[devout snow]

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[steel sun]

multiply everything by 25 and simplify

[steel sun]

multipled everthing by 25

[polar chasm]

good start

what did you get?

[steel sun]

lol

[steel sun]

d1+100-d1/125

[polar chasm]

huh

[steel sun]

i think im wrong

[woven radish]

MethIsAlwaysRight

[steel sun]

do i have to multiply the numerator aswell?

[polar chasm]

25 is 25 / 1

[steel sun]

ohhh

[polar chasm]

$=d_{1}+\frac{25}{1}\cdot\frac{100-d_{1}}{5}$

[woven radish]

MethIsAlwaysRight

[steel sun]

hell nah when did it become this big

[polar chasm]

note that 25 can also be written as 5 * 5, so one 5 cancels

[steel sun]

ohh

so d1+2500-d1/5?

[polar chasm]

note that the numerator would be 25 * (100 - d1)

so 25 * 100 - 25 * d1

[steel sun]

oh

[polar chasm]

but before doing this

[steel sun]

dumb me

[polar chasm]

we can cancel one 5

[steel sun]

yes

[polar chasm]

like this

then it becomes just 5(100-d1)

(so in total d1 + 5(100-d1))

[steel sun]

the answer given is this

[polar chasm]

mhm

it's correct

[steel sun]

how

ohhh

i get it

[polar chasm]

$=d_{1}+25\cdot\frac{100-d_{1}}{5}=d_{1}+\frac{5\cdot5}{1}\cdot\frac{100-d_{1}}{5}=d_{1}+5\cdot\left(100-d_{1}\right)=d_{1}+500-5d_{1}=500-4d_{1}$

this is the full algebraic manipulation that happened there

[woven radish]

MethIsAlwaysRight

[steel sun]

damn how did u learn to type that

[polar chasm]

tbf i type it on https://desmos.com/calculator

and then just copy paste it here

but it's not that hard

multiplication dot is \cdot and fractions are \frac

thats all that's needed for basic algebra

[steel sun]

oh

ok

.close

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[devout snow]

Please don't occupy multiple help channels.

[tired scarab]

Questions 9 to 10

i dont get it

[devout snow]

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[tired scarab]

<@&286206848099549185>

.close

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[devout snow]

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[steel sun]

i used the fact that for a quadratic to have integer solutions, the discriminant should be a perfect square

and i got k = 16 for the smallest square number

however, im still struggling to find N cuz i cant see any other way than trial and error, which will be lengthy

[sacred hound]

Is that true?

I think you can only have rational, not necessarily integer

[steel sun]

i may be wrong

[sacred hound]

But you can use that fact to find integer solution knowing that discriminant term is integer

[steel sun]

wdym

[sacred hound]

like, -b+-k is divisible by 2a

because discriminant is perfect square, k is integer

[steel sun]

yes

ohh

[tall stirrup]

k = \sqrt(240 + D^2)

D ∈ Z

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[wary ruin]

I'm currently working on a problem where I need to draw the function f(x,y) = x² + y² - 1. I've managed to figure out that the level curves of this function are circles with different radii. Specifically, for a given level curve, the radius is √(c+1), derived from the equation x² + y² = c + 1. However, I'm struggling with how to actually draw these level curves accurately. Could someone guide me through the steps or provide a method to visualize these circles for different values of c? Any help or tips would be greatly appreciated!

[lusty sapphire]

Draw a circle of radius 1, and label it z=0, draw a circle of radius sqrt(2) and label it z=1, draw a circle of radius sqrt(3) and label it z=2, so on

[wary ruin]

Is this for drawing the level curves?

[lusty sapphire]

yes

[wary ruin]

I have done that already, I need help visualizing the function in 3D.

[lusty sapphire]

You're given the z for each curve. That's your height

[wary ruin]

So I draw circles around the z-axis?

[lusty sapphire]

can you show me what you are trying to do?

[wary ruin]

@lusty sapphire

I’ve marked that it starts from z = -1.

Ignore the level curves right above, since I mistakenly drew the coordinate system too close…

[lusty sapphire]

Yeah that's pretty much right

[wary ruin]

I checked geogebra and it gave me the same

How strict do you have to be with the points (x, y, z) on the plane?

I only did a rough sketch because I am not used to, nor good at, drawing

@lusty sapphire

[lusty sapphire]

rough sketch is fine

As long as one can tell what you are doing

[wary ruin]

Ok thanks

.close

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[restive river]

hey guys

[restive river]

how did they find g(x) and p(y)

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[umbral raven]

I don't know how to solve this

[umbral raven]

I'm rly confused

The question: "What is the area of the regular polygon? Round your answers to two decimal places."

Please ping me if you can help me

[lusty sapphire]

Find CD using sohcahtoa

[umbral raven]

So I know angle DCB is 36 right

okay so

9cos(36) = x

to find it right

@lusty sapphire

[lusty sapphire]

yes

[umbral raven]

so X = 7.281

but then i still gotta find what DB is

so lets say thats y

so 7.281tan(36)

which is

5.289

[lusty sapphire]

looks good

[umbral raven]

and the length of AB is 5.289*2

so about 10.578

and then the formula is

1/2 * P * A so

1/2 * 52.89 * 7.281

(385.09209)/2

192.546045

and then rounded to two decimal points is: 192.55

[lusty sapphire]

looks good

[umbral raven]

thx

can u help with one more

i just need u to check if theres any mistakes

Assume the triangle below is an equilateral triangle and the 9m segment is the radius of the circle which is perpendicular to the side of the triangle.

And this is my work:

Area of the circle: 81π

Each side of the triangle is 18

The height of the triangle is 9√3

(9√3 * 18)/2 = (162√3)/2 = 81√3

**81√3 - 81π
**

wait no it'd be: 81π - 81√3

cause ur subtracting the circle from the triangle

[lusty sapphire]

How did you get side length is 18?

[umbral raven]

cause it says the triangle is eqilateral and the raidus of the circle is perpendicual to the side of the triangle

which means so is the diamater

and diamater is 9*2 = 18

[lusty sapphire]

well yeah diameter is 18, but how does that mean the side length of the triangle is 18?

[umbral raven]

Assume the triangle below is an equilateral triangle and the 9m segment is the radius of the circle which is perpendicular to the side of the triangle.

isn't that it what it means

[lusty sapphire]

This means they meet at right angles. It does not mean the sides are equal

[umbral raven]

oh

then what would i do

[lusty sapphire]

Find CE using sohcahtoa

[restive river]

Just use Pythagoras since it's an equilateral triangle

[umbral raven]

Is that triangle equilateral

[restive river]

.

[umbral raven]

But the little triangle

EDC

[lusty sapphire]

yes

[umbral raven]

so that means CE is just 9

wait no

it's a right triangle

so CE is 9√3?

Cause it's a 30-60-90 triangle

[fathom bridge]

Ce is 9(sqrt(3))

[umbral raven]

so it is 9√3

[fathom bridge]

Ya

cot(30) 9

[lusty sapphire]

Not sqrt(2). No

[fathom bridge]

…

[umbral raven]

I mean sqrt 3

[lusty sapphire]

yes

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[restive river]

I would love some help with this probability problem and just some tips and ways to solve problems like thiese. I just have no idea what to do

1. A fair die is rolled three times in a row.
   (a) What is the probability that a six was rolled first, given that the sum of all three rolls is 10?
   (b) What is the probability that a one was rolled at least once, given that the sum of all three rolls is 10?
   (c) What is the probability that exactly two sixes were rolled, given that at least one six was rolled?

[bleak breach]

B is 0

Wait nvm

I’m stupid

[restive river]

hmm isn't there like a lot of cases then. would 2,4,4 and 4,4,2 count as separate cases or do we only count a case for different number combinations or does the change of order also add a case

for example the combinations that start with 1:

(1,3,6) (1,4,5).. would (1,6,3) and (1,5,4) also be valid or do we treat them the same as the ones before

[devout snow]

@restive river Has your question been resolved?

[restive river]

alright well assuming that now i was able to do a and b, but now i have no idea how to go about solving the c) part of the question.

What is the probability that exactly two sixes were rolled, given that at least one six was rolled? so all combinations are 6^3 which is 216 and assuming one is guaranteed to be 6 we only need 2 more which is 36 different combinations now but how can we check if only one of those remaining ones is 6..

<@&286206848099549185>

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@restive river Has your question been resolved?

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[restive river]

Guys I don’t really understand how the curvature defintion tells us about a Curve’s curvature. The defintion says curvature K=|dT/ds| where T is the unit tangent vector and s is the arc length parameter/function

[restive river]

( I have really bad internet so it might take me a minute to see your message)

[urban harbor]

I guess intuitively, if the tangent vector to a curve is changing directions, then there is curvature happening. If your curve is simply a straight line then the tangent vector is constant as you travel along it (no curvature). The more extreme the curvature, the more quickly the direction of the tangent vector is changing over a short distance (or small arc length), which is where the ratio with the arc length comes in

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[gaunt galleon]

hi y'all, i am trying to write $\exists ! x \in X : A_x$ where $A_x$ is some proposition with "naked" quantifiers, this is what i've got: $$\exists x : ( x \in X \land \forall y : (y \in X \implies (A_x \iff x=y)))$$ is this correct? I couldn't find much information about this online.

[woven radish]

belabutter

[gaunt escarp]

this seems correct

[gaunt galleon]

ok good lol

[gaunt escarp]

i think you could also have done something akin to

for all y in X(y!=x) not A_x

sorry too lazy to type out the latex lmao

[devout snow]

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[gaunt galleon]

so $$\exists x : (x \in X \land \forall y : (y \in X \implies (x \neq y \implies \neg A_x)))$$ ?

[woven radish]

belabutter

[gaunt escarp]

yea

another (possibly simplier) way that i just thought of could be

[gaunt galleon]

awesome

[gaunt escarp]

$\exists x \in X s.t (\forall y \in X, A_x \iff x=y)$

[woven radish]

Mar the Marey

[gaunt escarp]

if that makes sense

[gaunt galleon]

question: would it be acceptable to move the "exists y" to the front?

[gaunt galleon]

i am referring to this

[gaunt escarp]

do you mean the for all?

[gaunt galleon]

oh yeah, my mistake

[gaunt escarp]

all good

[gaunt galleon]

but yes

[gaunt escarp]

it depends

where are you thinking of moving it

[gaunt galleon]

after the x

[gaunt escarp]

yea you could

if you had something like

$\exists x \in X s.t. \forall y \cdots$

[woven radish]

Mar the Marey

[gaunt galleon]

thank you so much, this has been very helpful 🙂

[gaunt escarp]

yea

just be careful when using the existential quantifier and the universal quantifier together

as reversing the order does not maintain the meaning

$\exists x \in X s.t \forall y \neq \forall y \in X \exists x$

[woven radish]

Mar the Marey

[gaunt galleon]

yeah that makes sense

alright

thanks again

.close

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[restive river]

I need help with this

[restive river]

x^3 - 4x^2 - 7x + 10

[grizzled yew]

=0?

[restive river]

yes

[grizzled yew]

plug easy values in first

[restive river]

what are easy values

[grizzled yew]

-1,0,1

for cubics its better to a guess a root first

and then factor it out

[restive river]

I need to find the roots or sum like that

[grizzled yew]

yea

[restive river]

what do I do with this

[grizzled yew]

plug them in

as x and see if it becomes 0

[restive river]

okay okay wait

wait so

if it =0

what does that mean

[stable storm]

do you know the remainder theorem?

[restive river]

what

[grizzled yew]

its a root

[stable storm]

yeah basically that

[restive river]

okay so

what do I do with it

[stable storm]

long division afterwards

to factor

[grizzled yew]

what have you found?

[restive river]

1

[restive river]

whats that

[grizzled yew]

nice

now do (x^3-4x^2-7x+10)/(x-1)

to get a quadratic that you can solve

[restive river]

how do I di this

[grizzled yew]

what did you learn

in school

[restive river]

I dont know

I forgot

my exam is in 4 hours

[grizzled yew]

look up polynomial long division

[restive river]

read this

is this relevant

I found this and it looks similar

[grizzled yew]

yeah

[restive river]

the 2)

[grizzled yew]

do that

[restive river]

I dont know how

what does P(x) mean

<@&286206848099549185> help 😭😭😭😭

[outer merlin]

P(x) is just the function are you familiar with f(x)? Its the same things just different letter

[restive river]

ohhh okay yeah

so now what how do I do this

[restive river]

this

[outer merlin]

I would use polynomial divison

Do you know it?

[restive river]

no

I have no idea

[outer merlin]

Long Divison?

[restive river]

what that is

Im cooked

[outer merlin]

fr

[restive river]

help please bro 😭

my exam is in 4 hours

[outer merlin]

https://www.youtube.com/watch?v=U2f5VzKGNPk&ab_channel=BrianMcLogan

Learn that

[restive river]

but but theres a +10

and no x

What is the question?

[outer merlin]

Its the same principe

[restive river]

x^3-4x^2-7x+10/(x-1)

[restive river]

im gonna watch it

[outer merlin]

The video is only 4minutes take a look at it then give the problem a try

[restive river]

I dont understand anything he is saying

[restive river]

btw the 2) is similar

[frigid sleet]

Are you integrating this or just finding the partial fractions expansion

[restive river]

its what I did in class but I only dont understand

[restive river]

what do these words mean

Im tryna find the roots or something like that

[frigid sleet]

Oh okay

[outer merlin]

Thats why you need to divide with the known root then solve the quadratic as you normally would

[restive river]

if this helps

[restive river]

how do I divide it

please lord save me

factor theorem and long division

fun

IDK WHAT LONG DIVISION IS

or factor theorem

[restive river]

learn it bud

[restive river]

I dont understand

[outer merlin]

Through trial and error(luckily it being 1 in this case) gave you one of the roots X=1. This is why you are diving the given function with (x-1).

[restive river]

Aops

MY EXAL IS IN 4 HOURS BUDDY AND I HAVE 30 MORE QUESTIONS

[restive river]

yeah rhat I get it

[restive river]

where u put f(1) = 0, thats factor theorem

Go on art of problem solving

THE REST

[outer merlin]

Look at the vid and follow what he does

[restive river]

I DONT UNDERSTAND ANYTHING HE IS DOING

cooked

where did the 3x come from bro

in the vid

y’all are tricking me thats not what I have to do

I never did that in class

[restive river]

Do you speak english or french

The vid is english

this is whag my teacher did

[restive river]

french

[frigid sleet]

There is more than one method mila

[restive river]

Do you understand english

[restive river]

give me the other method

[restive river]

yes

can y’all explain the method my teacher did

it seems easier

[outer merlin]

What methos is that

[frigid sleet]

Show us

[restive river]

this 😭😭😭 2)

[restive river]

U would learn it in grade 3

righr

I have been strugling since midnight y’all

[outer merlin]

Where are you from?

[restive river]

belgium

[frigid sleet]

I wish my French was better I've only just got home from work ill try my best

[restive river]

the guy that was supposed to help me literally gave up and left

am I that bad at math

[restive river]

I can understand it if u explain it in english dw 😭

except maybe a few words

[outer merlin]

Dutch, German or French?

[restive river]

french

[outer merlin]

https://www.youtube.com/watch?v=6IugSJyv_Qk&ab_channel=MathématiquesBdeB

French. Watch it.

[restive river]

I never learned this

[outer merlin]

Watch the video and youll solve your problem

[frigid sleet]

Factorising, it's seems like the method you've shown above has some substation of values and checking equalities are satisfied. But I'm not sure I'm shattered @restive river

[outer merlin]

This one might be better: https://www.youtube.com/watch?v=6Nd3qxO0xGE&ab_channel=KhanAcademyFrancophone

[restive river]

I watched it

I dont understand

[frigid sleet]

What are you finding hard to understand

[restive river]

she lost me when she started subtracting

I give up y’all

I have another question tho 😔

x^3 - 8x^2 + 16x this I dont get it

[frigid sleet]

So when you subtract you get a polynomial of a lower degree. Than before

[restive river]

I give up

.close

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[mossy forge]

i am having a hard time understanding the solution to this problem: https://artofproblemsolving.com/wiki/index.php/2021_AMC_10B_Problems/Problem_18#Solution_7_.28Complementary_.2B_PIE.29

I tried to use complementary counting so it is equal to 1-P(not every even number appears before an odd number appears)=3/6 (chance for odd)+3/6(chance for even) times 3/6(chance for odd)+3/6(even) times 3/6(even) times 3/6 (odd) +3/6(even) times 3/6 (even) times 2/6 (even that is not the last even remaining) times 3/6 (odd) + same thing before times another 2/6 + etc... till infinity which I used geometric sum to find that the total sum of P is 9/8 which doesn't make sense as 1-9/8 is negative. I don't understand the solutions as at least the simpler ones don't seem to be taking account for the fact that an even number can appear infinite times before an odd number appears

[shut pagoda]

the simpler solutions are basically looking at only distinct numbers

after you roll a 4 you can ignore it

because it will have no effect on the final probability

after you roll a 4 there is an equal chance of the next distinct number being any of the remaining ones

[mossy forge]

wait i dont understand y it wont have any effect on the final probability casue what if u get a bunch of 4s with the 2 and 6 before u get the odd number

[shut pagoda]

i meant the final state

weather or not it satisfies the condition

sorry

[mossy forge]

oh

that kind of bends my brain

but it makes sense

.close

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[twin fjord]

Yes

It is x=10

[winter patrol]

x^2 - 20x = -100 when x=10

[twin fjord]

10, -100

[winter patrol]

x=10 is the x-coord of the lowest point
-100 is the y-coord

[twin fjord]

No

It is the same point

[winter patrol]

points on the xy plane are represetned by an x-coord and a y-coord

[twin fjord]

At x=10, there is the lowest point for which the function takes the value -100

[winter patrol]

wdym

this goes all the way back to coordinate geometry

points on the xy plane are represetned by an x-coord and a y-coord

click on the point at the vertex there

you should see (10,-100)

x=10 is the x-coord of the lowest point
-100 is the y-coord

do you know how to plot points?

or to read points from a graph

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[vital delta]

Given this series how to correctly determine the radius of convergence. I am in confusion because:
-If we use the formula (the inverse of d'Alembert Ratio Test) we get R = 1/2
-But I find the interval to be (-π/6, π/6) and somewhere I have read that the radius is the distance from the mid point which is π/6.
Although 1/2 is almost equal to π/6 it does not make sense getting two solutions

[devout snow]

@vital delta Has your question been resolved?

[vital crown]

Actually inverse d'Alembert Ratio Test also gives you the same answer

You'll have |2sinx| <= R, which when you solve for x gives the same interval of -pi/6 to pi/6

You probably forgot that it's sinx and thought of it as |2x| <= R

[vital delta]

Take a look how I get to the solution 1/2. Why is it wrong?

[vital crown]

It's because of a little detail in d'Alembert

Which you I think don't know

Basically when you doing your ratio test, you are ignoring sin(x)^n

However that would also be there

Normally we will have our x to be just a normal x

So even if you don't write it at the end

Like you write 2x <= R or 2 <= R, your R would be fine

But that's not always possible

[vital delta]

I deliberately missed out (sin x)^n because in my textbook that is how the formula for R is.
Using (sin x)^n helps me find the interval, that is how i knew it was (-π/6, π/6)

[vital crown]

Yes you should use it in the formula as well

I never was taught the formula with the name

But I used the same method

And this is always there

Because let's imagine it's not

Then the series 2^n/n^2 (sinx)^n
And 2^n/n^2 (100000x)^n
And 2^n/n^2 x^n
And 2^n/n^2 (x + 8)^n etc... would all have same interval

So it's important to also account for the x part

[vital delta]

Ok I understand that but wait

Using d'Alembert we get |2sinx|. But why do you say that it should be <= R and < 1. Isn't d'Alembert criteria if the limit is < 1 the series converges

[vital crown]

Yes my bad you're right <= 1, not <= R

[vital delta]

Yea but that way we are going to find x which is the interval and if apply the definition that the R is the distance from the center to end points it makes sense to be pi/6 but when I am using the formula in my textbook i get 1/2. That is why I am confused, maybe that formula doesn't apply in all cases

[vital crown]

Okay so

You will have |f(x)| <= 1

Then you solve this for x

You get |x| <= R

If f(x) = x, for example
Then R = 1

If f(x) = x - 1, then R = 2

And so on so forth

If f(x) = sinx, R = pi/2
If f(x) = 2sinx, R = pi/6

[vital delta]

Well if we solve for |sin x| < 1/2 in my problem we get (-π/6, π/6)

Then how do I determine the R

In this scenario it is a number like -R, R

But If I have gotten an interval like (-3, -1) how would I find that R = 1

[vital crown]

Here's the thing

sinx < 1/2

So solve for x to find R

x < arcsin(1/2)

Which is pi/6

[vital delta]

you mean sin x = 1/2, becuase sin x < 1/2 gives an interval?

[vital crown]

Actually it's sinx <= 1/2

So x <= arcsin(1/2) so |x| <= pi/6 = R

My phone is at 4% I have to go at any second now

[vital delta]

Ok no problem

[vital crown]

Hope you understood it, if not just reread the messages give yourself some time ai think you'll get what I' trying to convey

Best of luck, take care!

[vital delta]

Thanks

<@&286206848099549185> Can anyone explain why in one of the examples the formula provided in my textbook gives the correct radius of convergence but in the other it gives 1/2 when it should be π/6

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[toxic kettle]

idk

https://tenor.com/view/nope-baby-exit-bye-woops-gif-15207775

[vital delta]

It doesn't make sense the formula to work always i quess.
I should just coinsider the radius is the distance from the midpoint to center

.close

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[prisma gust]

Good to see a microsoft paint buddy lololol

[vital delta]

I tried that but either I didn't know how to use it or I need paid version idk

[devout snow]

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[raven swift]

help?

[restive river]

Do you know the relationship between radicals and exponents?

@raven swift

[raven swift]

sort of

[restive river]

Ok so I’ll write out how to do it on a piece of paper then send it and explain I guess

[raven swift]

thank you

[restive river]

I kinda wrote out as many steps as I could

I converted everything to exponent form first

I also wrote 16 and 8 out as 2^4 and 2^3, respectively

Exponents and radicals are pretty confusing at first so it might take a while to go through everything I did

[raven swift]

ok this is what i did 4x*x^2/3+2 x 3sqrtrootx^5

[restive river]

Uh it would be 4x • x^(2/3) + 2cbrt(x^5) so if you meant that x between the 2 and 3 to be a multiplication that’s correct

[raven swift]

whats cbrt?

[restive river]

After that you need to write 4x • x^(2/3) as 2x^(5/3) and 4cbrt(x^5) as 4x^(5/3)

It’s cube root

Sqrt = square root and cbrt = cube root

So like the 3rd root

[raven swift]

so how do i simplify this further?

[restive river]

You use the rule x^a • x^b = x^(a+b)

And the rule x^(b/a)= brt(x^b) where brt is the bth root

[raven swift]

4x*x^2/3=

whats x?

[restive river]

we don’t know what x is

Some value

[restive river]

Would be 4x^(1+2/3)

By the rule

[raven swift]

shouldnt it be 4x^2?

x on x^2/3

[restive river]

How do you get x^2

[raven swift]

4x*x^2/3=4x^2

[restive river]

Yea but how did you get that

You need to use the rule x^a • x^b = x^(a+b)

[raven swift]

but x is diff on both sides

we have 4x

and also x

[restive river]

Uh the 4 isn’t included in the thing being raised to the power

4x^a ≠ (4x)^a

Order of operations says we have to do exponents first

[raven swift]

so whats x on the right hand side?

4x?

[restive river]

Its 4x^5/3

Cuz

4x • x^2/3 = 4x^(1+2/3) = 4x^(5/3)

Also note x = x^1

[raven swift]

alr so after this we add it?

[restive river]

You add the exponents

Yes

I mean it’s harder to explain with fractions but let’s say we have x^a • x^b where a and b are natural numbers. That’s x multiplied by itself a times multiplied by x multiplied by itself b times

Overall we have x multiplied by itself a+b times hence x^(a+b)

[raven swift]

im refering to +cbrt8x^5

[restive river]

Oh

Yea you can add them because they are both x^5/3

Otherwise they couldn’t be added

[raven swift]

oh alr, i got 1

answer choice

[restive river]

Yea that’s right

[raven swift]

thx

.close

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[devout snow]

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[stable spear]

can someone help me here please

[light light]

The derivative by product rule is given by vu' + uv', which means: the second function x the derivative of the first + the first function x the derivative of the second

So you need to multiply z^3-z+2 by the derivative of 6-z^3

And then multiply 6-z^3 by the derivative of z^3-z+2

then add those things together

[stable spear]

-5z^4+21z^2-4z-6

is what i got

[light light]

For what

your answer?

[stable spear]

oh wait i simplified too much

i got the answer

(3z^2-1) in the first box

and 2nd box was (-2z)

[light light]

Yeah when you're using the product rule you generally dont have to simplify much

[stable spear]

this is part B

yeah

[stable spear]

7z^3-6z+12-z^5-2x^2

is what i got

ah thats wrong

i got -z^5+7x^3-2x^2-6z+12 and its wrong also

can someone help

nvm got it

part C

got it

.close

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[devout snow]

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[edgy pond]

hat kind of sign is here: (either positive sign, or negative sign, or positive definite, or negative definite) quadratic form.

[edgy pond]

<@&286206848099549185>

[safe jasper]

!15mins

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Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[safe jasper]

!original

[devout snow]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

@edgy pond Has your question been resolved?

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[dire slate]

Can someone here explain to be proof by contradiction

[dire slate]

$\neg r \rightarrow F \equiv r$

[woven radish]

TheKingPin

[dire slate]

I believe this is the idea

[final drift]

context?

[dire slate]

proving a proposition by solving its contradiction

i believe

i dont get it

[final drift]

okay proof by contradiction is basically a way to prove a statement S

we suppose S isn't correct

and then we run into a contradiction

thus our assumption was wrong

lets go through a basic example

"Prove that all primes other than 2 are odd"

Suppose that all primes other than 2 are not all odd

Thus, there exist an even prime p other than 2

However, then p/2 is an integer other than 1 and also a factor of p

thus p isn't prime, a contradiction

thus all primes other than 2 are odd

[dire slate]

so, we go with the assumption that the negation is true and try to solve it that way?

[gaunt galleon]

If you care about the logic: $A \implies B$ is equivalent to $$(A \land \neg B) \implies \neg A$$ You can prove this with truth tables.

[dire slate]

I dont think I follow

[woven radish]

belabutter

[gaunt galleon]

A implies B is what you want to prove. So you assume A and not B and prove that it implies not A. That is proof by contradition. If you fill out the truth tables for both expressions you'll find that they are equivalent.

[dire slate]

now Im defintely confused

im trying to think where im confused

so, here's my thought process.
You have your proposition $A \rightarrow B$

[woven radish]

TheKingPin

[dire slate]

for whatever reason you need to prove this by contradiction which instead of proving it true you prove it false

this proof type is nonsense

so how do you prove it false

[gaunt galleon]

you're still trying to prove that $A \implies B$ i true

[woven radish]

belabutter

[dire slate]

$A \rightarrow \neg B$

[woven radish]

TheKingPin

[dire slate]

that would be false

then just prove $A \implies B$

[woven radish]

TheKingPin

[dire slate]

why overcomplicate things

[gaunt galleon]

because given the situation one or the other comes more naturally

[dire slate]

fair enough

ok, so contradiction

we have to prove that false?

and fail

so prove that $A \implies B$ isnt true

[woven radish]

TheKingPin

[dire slate]

and fail the proof

[gaunt galleon]

yes, what is the negation of $A \implies B$?

[woven radish]

belabutter

[dire slate]

Well, I dont think theres an implication negation so you put it in the for $\neg A \lor B$

[woven radish]

TheKingPin

[gaunt galleon]

yes

[dire slate]

then, the negation is $A \land \neg B$

[woven radish]

TheKingPin

[gaunt galleon]

exactly

[dire slate]

so we try to prove this?

and fail

[gaunt galleon]

we lead this to a contradiction

so $(A \land \neg B) \implies \neg A$

[dire slate]

how, there's no way you can just know this will lead to a contradiction just by looking at it. Even if it does there's no way to know when you have reached a contradiction. People just assume their work is wrong

[woven radish]

belabutter

[dire slate]

wait what

how

[gaunt galleon]

you do this and this is logically equivalent to $A \implies B$

[woven radish]

belabutter

[gaunt galleon]

you assume A, which is presumably true and you also assume not B

[dire slate]

isnt that saying $(\neg (A \implies B) \implies \neg A)$ ?

[woven radish]

TheKingPin

[gaunt galleon]

if this implies not A, we are in trouble, because A should be true

[gaunt galleon]

yes

[dire slate]

im not sure i follow what the point of that is

why do we need to know that

[gaunt galleon]

so we must assume that B is true

[dire slate]

I thought the left side is what we assume true

[gaunt galleon]

know what?

[dire slate]

this

why not just stick to the negation of the original statement

and try to prove that

[gaunt galleon]

because when doing a proof by contradiction you don't think "not (A implies b)", you assume "A is true" and "not B is true" and then prove that this leads to a contradiction.

they are the same

but it's about what's more natural

[dire slate]

right, cause instead of proving that A implies B we prove its wrong, i.e !(a implies b) which is false, A implies B is only wrong when B is false

oh

so thats why we assume B is false

[gaunt galleon]

i think you get it

[dire slate]

ok, so we prove not (A implies B) true?

and find out its false'

[gaunt galleon]

we assume not (A implies B) and if it leads to a contradiction, we know it's false

therefore A implies B has to be true

[dire slate]

ok, so we go with the approach that not(A implies B) is a true statement

and as you said lead to a contradiction because its false

so it must be true that A implies B since not (A implies b ) is false

[gaunt galleon]

yup

[dire slate]

ok, i get that

how do you get a contradiction

or know you will get one

[gaunt galleon]

that is the "implies not A" part

you start with some assumption

that's A

and if not (A implies B) implies not A we have a contradiction

because A and not A can't be true at the same time

[dire slate]

im still confused with that not A approach, why not just say, assume A is true and B is false

be proven wrong so we assume A implies B is true

[gaunt galleon]

that's exactly what you do

[dire slate]

wait what

[gaunt galleon]

$(A \land \neg B) \implies \neg A$

[woven radish]

belabutter

[gaunt galleon]

assume A is true and B is false

[dire slate]

I get the left side is the negation (a implies b)

[gaunt galleon]

well if A is true and we somehow arrive at the conclusion that "not A" is also true, we have a contradiction

i think we need an example

[dire slate]

not sure, I follow, wouldn't that be $(A \land \neg B) \land \neg A$ ?

[woven radish]

TheKingPin

[gaunt galleon]

i don't think that would be logically the same as "A implies B" so no

[dire slate]

true but isnt that what were trying to prove

that A and not A comes out as true

which is a contradiction

[gaunt galleon]

you can't really prove that because it's always false

[dire slate]

right

thats a contradiction

[gaunt galleon]

for example: prove that a>b implies a+c>b+c

[dire slate]

Law of contradiction, thats what im thinking about $p \land \neg p \equiv \mathbf{F}$