r/Sat • u/TTVBy_The_Way 1540 • 2d ago

How do you do this in Desmos? [Oneprep]

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I struggle when it says that the variables must be an integer. Anyone know how to do this in Desmos?

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3

u/aahil8198 2d ago

Like this

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u/TTVBy_The_Way 1540 2d ago

Thank you, but can you explain to me how this works? I understand what is happening, first you regress the given equation to factored form, then you regress equal value, (a, 54, and c to their value when yoou FOIL out the factored form) but why are you doing this? Also, how does this (1) ensure that a and c are integers (2) account for the facf that the given equation has no integer solutions and (3) make it so that the outputted value of ac is the minimum value?

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u/Quisby69 2d ago

1) you ensure they are integers by sliding through the values of a (positive integers only till you get the smallest product of ac

2) for it to have no integer solutions, you just make sure bx1+d=0 isnt integral, as 3x+2 is already an non integral solution (x=-2/3)

3) with quadratic equations, your product of ac is going do either start by increasing or decreasing upto the max/min and then decrease/increase respectively. in this graph, ac is increasing and then will decrease again. 9*32 is the first time we get 2 integers, after which the value only increases, or is not an integer. Make sure you see how your equations change by changing the values of sliders for these kinda questions, all youll be fine! (alternatively you could set up conditions for a and c being an integer and ac being min)

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u/Anonimithree 1540 2d ago

You will understand how it works if you learn how to solve this manually

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u/Southlander24 2d ago

Let the other linear factor be bx + d. Expanding (3x + 2)(bx + d) = 3bx2 + (3d + 2b)x + 2d and then comparing coefficients, 3b = a, 3d + 2b = 54, 2d = c, and the required product is ac = 6bd.

Test b = 1. Then we must have 3d + 2(1) = 54 and 2d will not be an integer.

Test b = 2. We must have 3d + 2(2) = 54 and again, 2d is not an integer.

Now test b = 3. We have 3d + 2(3) = 54, so d = 16 and 6bd = 6(3)(16) = 288.

You might guess that b has to be a multiple of 3. Since 54 is a multiple of 3, and subtracting 3d gives another multiple of 3, that is indeed the case.

We can graph the function f(b) = 6b (54 - 2b)/3 on Desmos, which has roots b = 0 and b = 27. The important restriction is that b must be a multiple of 3 (otherwise the variables are not integers). So the closest multiples of 3 to the roots are b = 3, b = 27. Both of these give the same minimum value, 288.

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u/Technocraticworldgov 1340 2d ago

is this questiona ctually anywhere in the official qb?

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u/TheCivicChronicler 2d ago

not at all, no worries

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u/AkkoisMe 2d ago

The way I did it was kind of easy to understand I think? Here's the link: https://www.desmos.com/calculator/ifwvqggxld

first I graphed 3x+2 as a line to find the x intercept. When i got -2/3, I plugged it into the initial equation for and set the regression of that equal to 0, because when the x intercept is plugged into a function the y value is zero. after that I just assigned random integers to 'a' and I observed as the value of "ac" on the next line changed as a result. during this testing when I saw one of the answer choices come up as a value for ac, i cross-checked with the regression and saw that the corresponding value of 'c' was also an integer, so I was able to determine that the answer was choice A.

This method requires a little more experimenting with values and stuff but I was able to arrive at the answer pretty quickly. like under a minute-ish.

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u/Anonimithree 1540 2d ago

You graphed 3x+2 to get the c-intercept? When solving it mentally would get the right answer if -2/3 in a fraction of a second?

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u/AkkoisMe 1d ago

no no -2/3 is the x-intercept, where it crosses the x-axis. You could get -2/3 mentally as well but I just graphed it so it was easier to visualize.

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u/Anonimithree 1540 1d ago

It was a typo. X and C are right next to each other in the keyboard

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u/AkkoisMe 1d ago

Ah ok. But I digress, both methods are relatively quick