jfan17

Oh sorry, I saw your question as q(3) = 30, not -30. I already showed you how I would've gotten it with f(3) = 30, so can you build off of that?

First, we observe that if the function \(q(x)\) has a hole at x = -2, that would indicate that both the values of the numerator AND denominator are 0. (0/0 produces the hole in the graph). The important takeway here is that the function \(q(x)\) then has a root of -2

If the equation \(q(x)\) has a vertical asymptote at x = 1, that would indicate that the solution present at 1 is undefined. We know from this that 1 is a root of \(q(x)\). 

Since we have 2 roots of \(q(x)\) let's assume it's a quadratic with two roots(anyone have a more rigorous proof than this?). We have the general form as:

ax²+bx+c = 0

We know that -2 and 1 are roots of this quadratic, so by vietas formulas:

-2 + 1 = -b/a

-1 = -b/a

-a = -b

a = b

-2 * 1 = c /a 

-2 = c/a

c = -2a

(we will use these later :P)

Next we look at the final thing the problem gives us, which is that \(f(3) = 30\)

That means that substituting into the expression, we get:

9a + 3b + c = 30

Here's where our previous expressions come in handy:

We can express everything in terms of b here. We then get:

9(b) + 3b + c = 30

c = -2a

a = b

c = -2b

9b + 3b -2b = 03

10b = 30

b = 3

Since a = b

a = 3

since c = -2b:

c = -6

we then have the quadratic:

3x²+3x-6

on the bottom

I know, I'm just saying "if". However, the equations are formatted the exact same way(the amount of spacing is the exact same which caught my attention), with the logic and steps being the same? I'm not too sure.....

Not sure if this is a coincidence, but CTG's explanation goes exactly in tandem with Cphill's previous explanation of this problem. If you want to see for yourself, the original link is here:

https://web2.0calc.com/questions/pls-help-due-tmrow_10

Now that I take a closer look, it seems that CTG's explanation is the exact same as Cphills. @CalTheGreat if you're going to use Cphill as a reference, I'd recommend linking his explanation first and foremost! Again, this is just a big "if"

No prob :)

If you want more reference info, search up "stars and bars combinatorics" on the internet. That's the formula that CPhill used in order to reach his answer for the second problem. Stars and bars only works when the "balls" or things you want to distribute are indistinguishable, and the "boxes" or groups you want to divide those things into, are distinguishable(if it was flipped around, you need to use stirling numbers like Cphill used for the first question). 

Here's a wikipedia link to it, if you really want to learn more:

https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29

So, what exactly is the length CD? There is no point "C" ever mentioned within the problem itself.

\(f(x)=g(x)\)

The problem already gives us the respective expressions for these 2 functions. We can rewrite this as:

-2x = x²-3

Adding 2x on both sides, get :

0 = x²+2x-3

we can factor this into:

(x+3) * (x-1) = 0

Our roots for this quadratic are then:

x = -3

x = 1

I don't have too much experience with doing graphing letters specifically, but I'd imagine using parametric equations would make the job substantially easier. Maybe complex too?