In need of help with basic algebra. | Mathematics | Page 1

shell kraken Jun 5, 2023, 5:37 AM

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Unable to find the smallest a+b, in need of step-by-step guide and subject name (So I can watch lessons)

silver steppe Jun 5, 2023, 5:38 AM

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shell kraken Unable to find the smallest a+b, in need of step-by-step guide and subject name ...

is this 45 (decimal point) (concatenate that with a) or 45 times a

shell kraken Jun 5, 2023, 5:38 AM

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45 times a

silver steppe Jun 5, 2023, 5:38 AM

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so 45 is 5*9

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wait

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none of the answers are right

shell kraken Jun 5, 2023, 5:39 AM

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how would you solve it?

silver steppe Jun 5, 2023, 5:39 AM

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well b^2 >= 0 and so a >= 0 and wowwowwow a = 0, b = 0 is a solution

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OH

shell kraken Jun 5, 2023, 5:40 AM

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Need to go smaller than that, presumably

silver steppe Jun 5, 2023, 5:40 AM

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ok maybe the problem is right

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yeah

silver steppe Jun 5, 2023, 5:41 AM

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silver steppe so 45 is 5*9

if 45a is a perfect square, then 5a is a perfect square

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or perhaps more usefully, a/5 is a perfect square

shell kraken Jun 5, 2023, 5:42 AM

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What does this mean, and where do we go from here?

silver steppe Jun 5, 2023, 5:42 AM

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then a must be 5*n^2 where n is an integer, right?

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then 45a = 225n^2 = b^2
and b gotta be negative

shell kraken Jun 5, 2023, 5:43 AM

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ahhhh, 45a is 3^2 x 5a^2

silver steppe Jun 5, 2023, 5:44 AM

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yep

shell kraken Jun 5, 2023, 5:44 AM

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and from there...

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uhm.

silver steppe Jun 5, 2023, 5:45 AM

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silver steppe then 45a = 225n^2 = b^2 and b gotta be negative

so b = -15n

shell kraken Jun 5, 2023, 5:45 AM

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I'm stuck again...

silver steppe Jun 5, 2023, 5:45 AM

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a = 5n^2

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a + b = 5n^2 - 15n

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simple minimization

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problem

shell kraken Jun 5, 2023, 5:45 AM

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shell kraken ahhhh, 45a is 3^2 x 5a^2

I'm stuck back at this, what's the next step?

silver steppe Jun 5, 2023, 5:46 AM

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silver steppe then a must be 5*n^2 where n is an integer, right?

^

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since 45 = 3^2 * 5 * a, a must have a 5 in its prime factorization to make that 5 into a 25,

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and the rest of the factors pair up

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since b is an integer, so 45a is a perfect square

shell kraken Jun 5, 2023, 5:48 AM

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ah, uh

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i think i almost got it, just need to think more

shell kraken Jun 5, 2023, 5:51 AM

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silver steppe since 45 = 3^2 * 5 * a, a must have a 5 in its prime factorization to make that ...

wait, why is this?

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why must it become a 5^2?

silver steppe Jun 5, 2023, 5:51 AM

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like lets say a isnt divisible by 5

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well then 45a has exactly 5^1 in its prime factorization

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and it cant be a perfect square

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since all the prime powers need to be even

shell kraken Jun 5, 2023, 5:52 AM

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so for 45a to be equal to a square of an integer, it needs to be written as x^2 x y^2 x z^2 etc?

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the squares of other numbers give us squares?

silver steppe Jun 5, 2023, 5:53 AM

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sure

shell kraken Jun 5, 2023, 5:53 AM

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if a is a multiple of 5

silver steppe Jun 5, 2023, 5:53 AM

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in fact, a needs to have an odd number of 5's in its prime factorization

shell kraken Jun 5, 2023, 5:53 AM

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why

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because there's already a 5, and a 5 would let us write it as a square by becoming even?

silver steppe Jun 5, 2023, 5:54 AM

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because 45a needs an even number of 5s in its prime factorization

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and the 45 has 1

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so the a needs an odd number

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maybe this is not a helpful way to understand

shell kraken Jun 5, 2023, 5:55 AM

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so to be equal to the square of an integer, 45a needs to be written as x^2

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for us to convert 45 into a x^2, we need another 5

silver steppe Jun 5, 2023, 5:55 AM

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shell kraken so to be equal to the square of an integer, 45a needs to be written as x^2

well it's actually b^2 but ye

shell kraken Jun 5, 2023, 5:55 AM

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so a is a multiple of 5

silver steppe Jun 5, 2023, 5:56 AM

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shell kraken for us to convert 45 into a x^2, we need another 5

ye

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in fact, not just a multiple of 5, but 5 times a perfect square

shell kraken Jun 5, 2023, 5:56 AM

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so if a has one 5, we can convert the 45 into:
3^2 x 5^2 x (a/5)

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so 15^2 x a/5

silver steppe Jun 5, 2023, 5:57 AM

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mhm

shell kraken Jun 5, 2023, 5:57 AM

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and for a/5 x 15^2 to be the square of an integer, a/5 needs to be a square of something

silver steppe Jun 5, 2023, 5:57 AM

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correct

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i chose it as n^2

shell kraken Jun 5, 2023, 5:57 AM

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where do we go from here?

silver steppe Jun 5, 2023, 5:58 AM

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silver steppe i chose it as n^2

write everything in terms of that

shell kraken Jun 5, 2023, 5:58 AM

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15n^2 = b^2

silver steppe Jun 5, 2023, 5:58 AM

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225n^2

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= b^2

shell kraken Jun 5, 2023, 5:58 AM

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whuh

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where did the x15 come from?

silver steppe Jun 5, 2023, 5:59 AM

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shell kraken so 15^2 x a/5

15^2

shell kraken Jun 5, 2023, 5:59 AM

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but 15^2 x n^2 = 15n^2?

silver steppe Jun 5, 2023, 5:59 AM

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no

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you need the parentheses

shell kraken Jun 5, 2023, 5:59 AM

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if both are squares you multiply the bases?

silver steppe Jun 5, 2023, 5:59 AM

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(15n)^2

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is that what you meant

shell kraken Jun 5, 2023, 5:59 AM

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no

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like when you multiply 3^2 and 5^2 it becomes 15^2

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you multiplied the bases

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so here you multiply 15 and n

silver steppe Jun 5, 2023, 6:01 AM

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15^2 times n^2 is (15n)^2, no?

shell kraken Jun 5, 2023, 6:01 AM

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15n^2

silver steppe Jun 5, 2023, 6:01 AM

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yes but when you write that, it means (15)(n^2)

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due to order of operations

shell kraken Jun 5, 2023, 6:01 AM

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ah

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when you process 15n^2

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you get (15x15) x (nxn)

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225n^2 = b^2

silver steppe Jun 5, 2023, 6:02 AM

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so then you can get b in terms of n

shell kraken Jun 5, 2023, 6:03 AM

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by taking the square root of both, therefore b = 15n?

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n is a/5 so b = 3a?

silver steppe Jun 5, 2023, 6:03 AM

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n^2 is a/5

shell kraken Jun 5, 2023, 6:03 AM

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a

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225/5a = b^2

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45a = b^2...

silver steppe Jun 5, 2023, 6:05 AM

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just gets you back

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heres what i would do
a/5 = n^2 so a = 5n^2
b = 15n
so, a + b = 5n^2 + 15n
now you want to make that as small as possible

shell kraken Jun 5, 2023, 6:05 AM

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aaah

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n^2 = a/5 and b = 15n

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multiply the a/5 side by 5

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5n^2 = a

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5n^2 + 15n = a + b

silver steppe Jun 5, 2023, 6:07 AM

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thats the thing it asks about

shell kraken Jun 5, 2023, 6:07 AM

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I kinda get it now, will need practice though

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so the lowest value possible of 5n^2 +15n without bothering to find the a+b

silver steppe Jun 5, 2023, 6:07 AM

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mhm

shell kraken Jun 5, 2023, 6:07 AM

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there was a way of doing this, but I forgot

silver steppe Jun 5, 2023, 6:08 AM

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well, it's a parabola, so find the vertex and look for the nearest integer x value

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since the vertex is a minimum but it may not have integer

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and n must be integer

shell kraken Jun 5, 2023, 6:10 AM

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silver steppe well, it's a parabola, so find the vertex and look for the nearest integer x val...

I'm sorry for asking so many questions but how does one do this 💀

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I forgot so much about math

silver steppe Jun 5, 2023, 6:10 AM

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is graphing calc allowed lol

shell kraken Jun 5, 2023, 6:10 AM

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no calc allowed

silver steppe Jun 5, 2023, 6:11 AM

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so
factor it
you can find the 0s
x coord of vertex is the average of the 0s

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alternatively there is -b/2a

fervent imp Jun 5, 2023, 6:12 AM

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shell kraken Unable to find the smallest a+b, in need of step-by-step guide and subject name ...

We have 45a = b^2, so a = 45/b^2.
a needs to be an integer. So, factor 45 and try to see what values of b you can take so 45/b^2 is an integer.

shell kraken Jun 5, 2023, 6:12 AM

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ah, that formula!

shell kraken Jun 5, 2023, 6:12 AM

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silver steppe alternatively there is -b/2a

15/5 = 3

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what... is the result of this formula

silver steppe Jun 5, 2023, 6:13 AM

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it's 3/2,

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this gives the x value of the axis of symmetry

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or the x value of the vertex

silver steppe Jun 5, 2023, 6:13 AM

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silver steppe it's 3/2,

why did i say this
it's -3/2

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-15/(2*5)

shell kraken Jun 5, 2023, 6:14 AM

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-b/2a...
5n^2 - 15n

(-15) / 10?

silver steppe Jun 5, 2023, 6:14 AM

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yes right

fervent imp Jun 5, 2023, 6:14 AM

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Wait, wait. There aren't any parabolas here.

silver steppe Jun 5, 2023, 6:14 AM

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so to make this as small as possible we would like n to be -3/2

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however

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that is not allowed

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as it's not an integer

fervent imp Jun 5, 2023, 6:14 AM

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Hpw are you trying to solve this problem?

silver steppe Jun 5, 2023, 6:14 AM

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we are trying to minimize a + b

shell kraken Jun 5, 2023, 6:15 AM

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and a+b is 5n^2 + 15n

silver steppe Jun 5, 2023, 6:15 AM

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5n^2 + 15n

shell kraken Jun 5, 2023, 6:15 AM

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wait why is it a plus

fervent imp Jun 5, 2023, 6:15 AM

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Well, it's actually just easier to find the values of b that even produce integers, really.

silver steppe Jun 5, 2023, 6:15 AM

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shell kraken so the lowest value possible of 5n^2 +15n without bothering to find the a+b

because

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we agreed on that

silver steppe Jun 5, 2023, 6:15 AM

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fervent imp Well, it's actually just easier to find the values of b that even produce intege...

shush

shell kraken Jun 5, 2023, 6:15 AM

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ah, I misread our past convo

fervent imp Jun 5, 2023, 6:15 AM

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There aren't many.

silver steppe Jun 5, 2023, 6:16 AM

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rrrrr

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there are an infinite amount

fervent imp Jun 5, 2023, 6:16 AM

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No.

silver steppe Jun 5, 2023, 6:16 AM

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any multiple of 15 works

fervent imp Jun 5, 2023, 6:17 AM

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No. If 45/b^2 is an integer and b is also an integer, then there are only 4 values of b that work.

silver steppe Jun 5, 2023, 6:17 AM

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silver steppe so to make this as small as possible we would like n to be -3/2

so anyway we choose the closest value and evaluate the function

silver steppe Jun 5, 2023, 6:17 AM

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fervent imp No. If 45/b^2 is an integer and b is also an integer, then there are only 4 valu...

45/b^2 does not have to be an integer though

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that's 1/a

fervent imp Jun 5, 2023, 6:18 AM

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silver steppe 45/b^2 does not have to be an integer though

Ah, sorry! I slightly misread the problem. Let me think about it some more, then.

silver steppe Jun 5, 2023, 6:18 AM

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op come back :((((

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maybe they solved it

shell kraken Jun 5, 2023, 6:19 AM

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sorry, I haven't solved it, but I'm just

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not feeling great so I took some meds

fervent imp Jun 5, 2023, 6:20 AM

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So, we have a + b = b^2/45 + b.
As 45 = 3^2*5, we must have b = 15n for b^2/45 to be an integer.
Thus, we are minimizing f(b) = (15n)^2/45 + 15n = 5n^2 + 15n.

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Aha! Now I see your strategy.

silver steppe Jun 5, 2023, 6:20 AM

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shell kraken not feeling great so I took some meds

oh rip

fervent imp Jun 5, 2023, 6:21 AM

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shell kraken not feeling great so I took some meds

Yeah, take a break! We can discuss it later. Rest is important.

silver steppe Jun 5, 2023, 6:21 AM

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good idea

shell kraken Jun 5, 2023, 6:21 AM

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no, i'd rather finish this then rest

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otherwise it will plague me the whole time

silver steppe Jun 5, 2023, 6:21 AM

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so we established that the minimum of 5n^2 + 15n is when n = -3/2 but this is not allowed

shell kraken Jun 5, 2023, 6:22 AM

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and we got that from -b/2a, which is the formula we use for parabolic equations to find the...

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the...

silver steppe Jun 5, 2023, 6:22 AM

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the horizontal position of the vertex

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the x value of the axis of symmetry

shell kraken Jun 5, 2023, 6:22 AM

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the x of the vertex

silver steppe Jun 5, 2023, 6:22 AM

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or in this case ~~the n value~~

shell kraken Jun 5, 2023, 6:22 AM

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hmm, so 5n^2 + 15n = 0?

silver steppe Jun 5, 2023, 6:23 AM

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no

shell kraken Jun 5, 2023, 6:23 AM

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Did I confuse it with another type of equation?

silver steppe Jun 5, 2023, 6:23 AM

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since the vertex isn't allowed choose the closest integer value of n

silver steppe Jun 5, 2023, 6:23 AM

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silver steppe since the vertex isn't allowed choose the closest integer value of n

which will still be pretty close to a minimal point of the parabola

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as it's close to the vertex

fervent imp Jun 5, 2023, 6:25 AM

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We should compare the closest integer values to the vertex. After all, the parabola has the smallest value at the vertex and inreases away from it.

shell kraken Jun 5, 2023, 6:29 AM

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What... is the next step?

silver steppe Jun 5, 2023, 6:29 AM

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fervent imp We should compare the closest integer values to the vertex. After all, the parab...

e

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this is general solution to minimize quadratic over the integers

fervent imp Jun 5, 2023, 6:30 AM

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Well, we got the result that if any n was allowed, then the minimum of f(n) = 5n^2 + 15n would be at n = -3/2.
But we need integer n. So, let's instead substitute the two integer values of n closest to -3/2.

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What will they be?

shell kraken Jun 5, 2023, 6:33 AM

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-1?

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wait, let's try -2

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-20 + -30

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-50

fervent imp Jun 5, 2023, 6:36 AM

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-20 isn't correct.

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If n = -2, what is 5n^2?

shell kraken Jun 5, 2023, 6:37 AM

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is it (5n)^2?

fervent imp Jun 5, 2023, 6:37 AM

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No.

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Just 5n^2, as we discissed above.

shell kraken Jun 5, 2023, 6:37 AM

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ah 4

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-2 x -2

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therefore 20

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therefore -10

fervent imp Jun 5, 2023, 6:37 AM

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Yes.

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Right.

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And now try -1.

shell kraken Jun 5, 2023, 6:38 AM

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-1
5 -15
-10 again

fervent imp Jun 5, 2023, 6:38 AM

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Yes. Why do you think that happened?

shell kraken Jun 5, 2023, 6:38 AM

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...I don't know...

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this is the absolute lowest possible, therefore is the answer?

fervent imp Jun 5, 2023, 6:39 AM

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Yes.

shell kraken Jun 5, 2023, 6:39 AM

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So let me write the solution of this question step-by-step from memory with my own words

silver steppe Jun 5, 2023, 6:40 AM

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shell kraken ...I don't know...

it's because they are the same distance from the vertex, and parabolas are symmetric around the axis of symmetry
they are the same distance from that axis
so y value has to be the same

fervent imp Jun 5, 2023, 6:40 AM

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By the wy, the reason for these two values being equal is that the vertex at n = -3/2 is equidistant from n = -2 and n = -1. And as the parabola is symmetric around an axis passing through the vertex, these values have to be the same.

shell kraken Jun 5, 2023, 6:40 AM

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45a = b^2
therefore 45^a is n^2
for it to be written as a square, I need to convert 45 into x^2
3^2 x 5 is 45, so it needs another 5, therefore a needs to be divisible by 5 to steal from it, and 1/5 needs to be a square as well.
15^2 x a/5 = b^2

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a/5 is the square of something, so we integrate it as 15n^2 = b^2

fervent imp Jun 5, 2023, 6:41 AM

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Wait, wait. What does 45^a have to do with anything?

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If 45a = b^2, then a = b^2/45.

shell kraken Jun 5, 2023, 6:42 AM

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b = 15n

fervent imp Jun 5, 2023, 6:42 AM

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Yes, because 45 = 3^2*5.

shell kraken Jun 5, 2023, 6:43 AM

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shell kraken b = 15n

and n^2 = a/5 therefore a = 5n^2

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a+b = 5n^2 + 15n

fervent imp Jun 5, 2023, 6:44 AM

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Yes.

shell kraken Jun 5, 2023, 6:44 AM

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and we use -b/2a to find the x of the vertex of a parabola

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in which case it is -15/10

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-3/2

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but it needs to be an integer, so we have to try the closest integers to it

fervent imp Jun 5, 2023, 6:45 AM

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Yes.

shell kraken Jun 5, 2023, 6:45 AM

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first we tried -2 which gave us -10

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-1 did the same

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this proves that -10 is the lowest value of a+b

fervent imp Jun 5, 2023, 6:45 AM

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Correct.

shell kraken Jun 5, 2023, 6:45 AM

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and -10 is the answer of this question

silver steppe Jun 5, 2023, 6:45 AM

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🥳 hooray

shell kraken Jun 5, 2023, 6:46 AM

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equations in forms of ax^2 +- by +-c are paraboles

fervent imp Jun 5, 2023, 6:46 AM

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No.

silver steppe Jun 5, 2023, 6:46 AM

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y = ax^2 + bx + c

fervent imp Jun 5, 2023, 6:46 AM

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Yes.

silver steppe Jun 5, 2023, 6:46 AM

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in this case, it was an^2 + bn + 0

shell kraken Jun 5, 2023, 6:46 AM

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i forgott he ^2 whoops

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the c was 0

fervent imp Jun 5, 2023, 6:47 AM

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We can modify the problem a bit. Then you can solve it, just to make sure you got it.

shell kraken Jun 5, 2023, 6:47 AM

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so the things I learned from this question
things that are equal to the square of an integer have to be written in squares of integers so you can combine them to equal to the other side

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-b/2a is the x of a vertex of a parabole, the point at which y is lowest

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?

fervent imp Jun 5, 2023, 6:48 AM

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Well, it's the lowest if a > 0.

silver steppe Jun 5, 2023, 6:48 AM

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true

shell kraken Jun 5, 2023, 6:49 AM

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if a is < 0, then it is its highest point?

fervent imp Jun 5, 2023, 6:49 AM

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Yes.

shell kraken Jun 5, 2023, 6:51 AM

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fervent imp We can modify the problem a bit. Then you can solve it, just to make sure you go...

That would be delightful, thanks.

paper sapphireBOT Jun 5, 2023, 6:51 AM

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@shell kraken has given 1 rep to @fervent imp

shell kraken Jun 5, 2023, 6:52 AM

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by the way

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how about paraboles of those shapes?

fervent imp Jun 5, 2023, 6:58 AM

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shell kraken That would be delightful, thanks.

Now, suppose we have 12a = 5b^2 and we are trying to minimize a + 3b. a and b are integers, as before.

fervent imp Jun 5, 2023, 6:59 AM

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shell kraken

The ones in the first column can't be represented as y = f(x), as they have several values of y for some values of x. You can represent them as x = ay^2 + by + c, though.

shell kraken Jun 5, 2023, 7:00 AM

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ah so they're x = f(y)

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they're perpendicular to similar functions, in a way

shell kraken Jun 5, 2023, 7:00 AM

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fervent imp Now, suppose we have 12a = 5b^2 and we are trying to minimize a + 3b. a and b ar...

2^2 x 3 x a = 5b^2

lyric spruce Jun 5, 2023, 7:00 AM

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You could have just did, 5.3²a = b² ?

fervent imp Jun 5, 2023, 7:02 AM

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lyric spruce You could have just did, 5.3²a = b² ?

That's what we already discussed.

fervent imp Jun 5, 2023, 7:02 AM

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shell kraken 2^2 x 3 x a = 5b^2

Yes. So, what is a?

lyric spruce Jun 5, 2023, 7:03 AM

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I see

shell kraken Jun 5, 2023, 7:03 AM

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fervent imp Yes. So, what is a?

6^2 x a/3 = 5b^2

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say a/3 is n

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6n^2 = 5b^2

fervent imp Jun 5, 2023, 7:04 AM

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No, not a/3.

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Better to do it like this:
a = 5b^2/12 = 5b^2/(2^2*3)
So, we have 2 and 3 in the denominator. What should we take b as to get rid of those?

shell kraken Jun 5, 2023, 7:07 AM

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I'm lost...

fervent imp Jun 5, 2023, 7:07 AM

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Well, we need to take b = kn, where k is divisible by both 2 and 3.

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What is the smallest number divisible by 2 and 3?

shell kraken Jun 5, 2023, 7:08 AM

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6

fervent imp Jun 5, 2023, 7:09 AM

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Yes. Thus, k = 6 and we take b = 6n.

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So, now we look at a = 5b^2/12 again. Substitute b = 6n here.

shell kraken Jun 5, 2023, 7:09 AM

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5n^2 = a

fervent imp Jun 5, 2023, 7:10 AM

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No, not 5n^2. Calculate it more carefully.

shell kraken Jun 5, 2023, 7:10 AM

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180n^2/12

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15n^2

fervent imp Jun 5, 2023, 7:10 AM

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Yes.

shell kraken Jun 5, 2023, 7:10 AM

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15n^2 = a

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and b is 6n

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a + 3b is 15n^2 + 18n = y

fervent imp Jun 5, 2023, 7:11 AM

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So, we got that b = 6n, a = 15n^2.
Now, we return to the expression we want to minimize: a + 3b.

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Yes.

shell kraken Jun 5, 2023, 7:11 AM

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-b/2a time

fervent imp Jun 5, 2023, 7:11 AM

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Yes.

shell kraken Jun 5, 2023, 7:11 AM

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-15/12

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-5/4

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if we want the end result to be an integer, we take the closest ones to it

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-1 and -2

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starting with -1

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-3

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for -2, it is

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60 - 36

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24?

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help

fervent imp Jun 5, 2023, 7:13 AM

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shell kraken -15/12

Why 12?

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Our expression is 15n^2 + 18n.

shell kraken Jun 5, 2023, 7:14 AM

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ahhh

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calculation error again...

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-18/30

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-3/5

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where from here?

fervent imp Jun 5, 2023, 7:16 AM

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The value is the smallest at n = -3/5, but we need an integer n. What's the integer closest to -3/5?

shell kraken Jun 5, 2023, 7:17 AM

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-1

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from -1 we get -3

fervent imp Jun 5, 2023, 7:17 AM

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Yes. And that's the answer.

shell kraken Jun 5, 2023, 7:17 AM

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ahhhh

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how am I supposed to learn all this and become fluent in 3 months...

fervent imp Jun 5, 2023, 7:19 AM

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Don't worry, you just need some practice.

shell kraken Jun 5, 2023, 7:19 AM

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Yes... I'll get better at math and physics, and I'll get into medical school. Thank you very much.

fervent imp Jun 5, 2023, 7:20 AM

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You're welcome!

shell kraken Jun 5, 2023, 7:20 AM

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I am very thankful you bore with my slow-learner ass. I would be delighted to receive your assistance next time.

fervent imp Jun 5, 2023, 7:21 AM

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Well, don't worry! You'll certainly get better with some practice. Remember to take breaks, too.

#In need of help with basic algebra.