random question

[neon temple]

Find me two numbers such that when they are added together, they make as much as the cube of the lesser added to the product of its triple with the square of the greater; and the cube of the greater added to its triple times the square of the lesser makes 64
more than the sum of these two numbers.

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[gray geode]

let the larger no. =x

smaller = y

just type it in this form lol

[neon temple]

hm?

[gray geode]

algeabric

[neon temple]

what do we get though

[gray geode]

just write it first

[lofty quarry]

what anonhuman said; write the smaller number as y and the larger as x

then represent the numbers using equations

although i would like to question whether you need help with this question or simply wish to test us plebian helpers

[neon temple]

just felt like having someones input on this

and how people would solve for either a or b

[gray geode]

factorisation?

[neon temple]

how would you factor something like that?

[gray geode]

let the x+y = a

if you equate them

and then subtract them

[lofty quarry]

so in algebra, what you’re asking is $x+y = y^3 + 3x^2y$ and $x^3 + 3xy^2 = x+y+64$ with $y<x$?

[willow flicker]

Micabo #teaisthebest

[neon temple]

yeah

[lofty quarry]

x, y are real numbers?

[neon temple]

yes

[lofty quarry]

x^3 - 3x^2y + 3xy^2 - x^3 = (x+y+64) - (x+y) = 64

(x-y)^3 = 64

x-y = 4

x = y+4

2y+4 = y^3 + 3y(y+4)^2

i don’t want to solve this

[lofty quarry]

2y+4 = y^3 + 3y^3+24y^2+48y

4y^3 + 24y^2 + 46y - 4 = 0

[gray geode]

id you notice, an identity may form

if you input value of x+y

[neon temple]

yeah because someone else did the same thing but added the two equations instead of eliminating x+y

[lofty quarry]

i’m not solving this without a calculator but y is positive and approximately 0

[neon temple]

so they had to solve a depressed cubic in terms of x+y

[gray geode]

we will get x-y=4

x=4+y

[lofty quarry]

so (x+y)^3 = 2(x+y) + 64

[gray geode]

put that in

[lofty quarry]

i already said this

[lofty quarry]

.

[neon temple]

+64

but yeah

[gray geode]

oh yeah

[lofty quarry]

typo; sorry

[neon temple]

yeah

[lofty quarry]

this has no rational solutions, so i am unable to continue from here

maybe you can finish with the cubic formula

[neon temple]

thats what we did

and of course his approach was longer

and then using the value of the cube root in the second cubic

[neon temple]

this is what i got so yeah it just made the whole problem easier to work with

[tranquil edge]

or add them together

2(x+y) + 64 = (x+y)^3 and then use x+y as a term of depressed cubic]

look up formula

and then find integer values

[lofty quarry]

there are no integer values

[tranquil edge]

there aren't

[tranquil edge]

for this one?

[lofty quarry]

for x and y

[tranquil edge]

any int for 4y^3 +24y^2+46y - 4 = 0 => 2y^3 + 12y^2 + 23y-2=0

[lofty quarry]

$$ x + y = x^3 + 3xy^2 $$
and
$$ y^3 + 3yx^2 = 64 + x + y $$
for
$$ x < y $$ and $$ x,y \in \mathbb{R} $$

you seem a bit late

also, that's incorrect

Is it solved

I think those should be the eqns

it's x+y+64 on the right of the second equation

Oh the more than sum of numbers line I didnt read

[willow flicker]

roltt

[lofty quarry]

Is it solved

pretty much

Whats the answer

x + y = 4

x is approximately 0.083287 and y is x+4

[lofty quarry]

actually, y-x = 4

it's a bit confusing because we set x to be the larger of the two variables

$$ 2(x+y) = (x+y)^3 - 4^3 $$

[willow flicker]

roltt

[lofty quarry]

Right?

and then we apply a^3 - b^3

I think

[lofty quarry]

won't help

Wont hlep yea

How else did we get it

consider reading the chat

Ok

[chilly geyser]

I have an idea

[chilly geyser]

Now we use x = y+4

$2(4+2y) = (4+2y)^3-4^3$

[willow flicker]

clonesolopros

[chilly geyser]

If $a^3 - b^3 = (a-b)(a^2+ab+b^2)$ then for $a=4+2y$ and $b=4 \$
$(4+2y)^3 - 4^3 = (4+2y-4)( (4+2y)^2 + 4(4+2y) + 4^2) \ = 2y( (4+2y)^2 + 4(4+2y) + 16)$

[willow flicker]

clonesolopros

[chilly geyser]

$$\implies 2(4+2y) = 2y((4+2y)^2 + 4(4+2y) + 16)\$$
$$\implies 2(4+2y)-32y = 2y(4+2y)(4+2y +4) $$

[willow flicker]

clonesolopros

[chilly geyser]

$$ 4(2-15y) = 8y(2+y)(4+y) $$

[willow flicker]

clonesolopros