Quazars

Thank you this helped me get the correct answer when plugging in values for $k_1, k_2$ and $v_f$

I still struggle to understand how did you solve to find these x(t) and v(t)?

This question was inspired by this post https://math.stackexchange.com/questions/20717/how-to-find-solutions-of-linear-diophantine-ax-by-c/20738#20738

I didn't find the explanation that helpful so I was hoping someone could clarify or even put it into a more systematic setup where it can be solved by a matrix inverse.

Start with the 4 "poles" i.e (5,0), (0,5), (-5,0), (0,-5) 

Each quadrant of the circle has 2 integer values (3,4) and (4,3) - Since (3,4,5) is a pythagorean triple

4 quadrants give 8 more points so 12 points in total

Can you do that even if the log is in base 10? As in do you not have to raise it to the power of 10 instead?

How do you go from step 2 to 3? As in how do you simplify e^((log(2) log(3))/(log(4))) to sqrt(3)?

(If someone could help me with the Latex images just a bit bigger that'd be great, not sure why but they come up a little tiny)

I managed to get the answer myself, posting it incase anyone is curious:

Start off by changing the log from base 10 to base 2 since 2 and 4 happen to give a integer power of 2

$\frac{\log_{2}{x}}{\log_{2}{10}}=\left( \frac{\log_{2}{2}}{\log_{2}{10}}\frac{\log_{2}{3}}{\log_{2}{10}} \right)\frac{\log_{2}{4}}{\log_{2}{10}}$

$\frac{\log_{2}{x}}{\log_{2}{10}}=\left( \frac{1}{\log_{2}{10}}\frac{\log_{2}{3}}{\log_{2}{10}} \frac{\log_{2}{10}}{2}\right)$

$\log_{2}{x}=\frac{\log_{2}{3}}{2}$

$x=2^{\frac{1}{2}\log_{2}{3}}$

$x=\sqrt{2^{\log_{2}{3}}}$

$x=\sqrt{3}$

Any other methods are very welcome

One way to think about this type of questions is to make sure that the units on both sides are equal. 

Consider the left hand side of the equation which is always velocity squared. Since [v] = m^1 * s^(-1) we know that the unit for v^2 is m^2 * s(-2)

Consider the right hand side, all of them have a force T which has the unit kg^1 * m^1 * s^(-2)

Now let's look closer at the available options. In order for the right hand side be equal to the left hand side we need to get rid of one unit of kg and get one more unit of m.

If you look at the options m happens to be the mass and  x the length. Therefore we can see that dividing by one m we can balance the kg part and by multiplying one x we will get an additional m(meters) in the equation. Therefore the answer C has to be correct, we can double check this by putting in the units of all values to check:

$\frac{\text{m}^2}{\text{s}^2}= \frac{\text{kg}^1 \space \text{m}^1}{\text{s}^2} \cdot \text{kg}^{-1} \cdot \text{m}^1$

We can see that it works out and gives us the correct units!

The square root means that you want to find the number that when which multipliced by itself gives you the original number. I'm not sure how to nicely phrase it in a sentence so it's probably easiest with examples!

$\sqrt{4} = 2 \space \space \text{because} \space \space 2\cdot 2 =4$

$\sqrt{9} = 3 \space \space \text{because} \space \space 3\cdot 3 =9$

$\sqrt{16} =4 \space \space \text{because} \space \space 4\cdot 4 =16$

$\sqrt{25} = 5 \space \space \text{because} \space \space 5 \cdot 5 = 25$

Hopefully you are starting to see the pattern of how this works! Not all square roots have nice answers however! For example look at the square root of 2. Which is an ugly and neverending decimal number. It's approximately equal to 1.41

$\sqrt{2} \approx 1.41421356237$

Not as nice as the others huh? Most numbers don't have nice and clean answer! 

If I understood correctly (the $ signs are cluttering up the syntax somewhat) you want to solve the following:

$\sqrt{7+\sqrt{4-\sqrt{3+x}}}=3$

We also know that there is only one solution, since you said "only value of x".

One way to solve this is to look at the most outer square root with the 7. We know that the square root of 9 is 3. The only possible way this can hold true is if 7 + the other value is equal to 2. Therefore this has to hold true:

$\sqrt{4-\sqrt{3+x}}=2$

By using the same logic, 4 + something has to be equal to 4 in order for its root to be equal to 2. Therefore we can see this has to be true:

$\sqrt{3+x}=0$

This is clearly true only if x = -3

You can verify this by putting that value back into the original question. The other more general way is to just square both sides and move the term from the left to the right and keep squareing til you get x alone on the left and a number on the right. This will give you the same result but I thought that the explanation above would give you a better intuition(I hope)

$F(n) = \sum_{i=1}^n 0.5*10^i$

This would be the n'th term in this sentence if I understood your question correctly. You can input n=1,n=2 to get the first and second term. By putting in n=1000 you will get your desired answer. I'm not going to type out a thousand 5's but hopefully you get the idea of why this works. 

Oh... is there no way to simplify that? It seems like a very messy answer. How about the second case with an i coefficient infront of the x^2 term? thanks