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Hello Guest! Hor are you? REspond please!

Hello! I managed to do it but my method was very messy. It involved some calculus, mostly geometry, but since im not that good at math, my page looked like a 3 year old had scribbled on it for 5 years after i got 6 as my answer!

There are lots of gaps in your question where LaTeX should have been. If this question is on alcumus, click on the LaTeX and copy paste it in, thank you!

So first, lets understand this problem with an equation. This is what i got:

\(\frac{x}{x+1}\cdot\frac{x+1}{x+2} ...\frac{x+19}{x+20}=3\)

now we cancel. You see by cross canceling, the x+1's dissapear as all of them do and you are left with

\(\frac{x}{x+20} = 3\)

well, you might think that since it has to be a positive fraction and x has to be negative, this is wrong, BUT keep in mind, if both x in the numerator and denominator or negative, the fraction is a postive. so simplify this equation.

\(x = 3x+60\)  multiply both sides by \(x+20\)

\(2x=-60\)   subtracting x on both sides and moving the 60 over.

you will get x = -30, so the first fraction will be\(\frac{30}{31}\),which will be your answer!

Im very confused. You say it is a triangle, but it is not. Do you mean connecting Q and R to form a triangle? But then the answer would be pretty straight forward. \(\frac{12*14}{2}\) which will be 84. But i assume you mean the area of the figure shown here. So i will solve that too

Here is what i hope to be the solution: 

First lets find \(\overline{PQN}\) which will be 42 (use the triangle calculation formula \(\frac{base*height}{2}\))

Heres where things get trickier, but I want you to figure it out, but i will give you some hints

1. draw a line from P to the midpoint of \(\overline{QR}\)

2. try to see if you can combine triangles in some way to single out \(\overline{NOR}\) to find it

Hint: since each one of them is less than 3, the only possible way is the stick being broken into length 2,2, and 2.

find the total ways the stick can be broken up into. then it is 1/(total ways) <---- that will be your answer.

Hope this helps!

The quadratic arranges to \(x^2 - 15x + 36 = 0\)

Then by the quadratic formula,

\(x = \frac{15 \pm \sqrt{15^2 - 4 \cdot 36}}{2} = \frac{15 \pm 9}{2} = 12, 3\)

Therefore, \(p - q = 12 - 3 = \boxed{9}\)

The roots are 3 and 5, so p - q = 5 - 3 = 2.