System of Laplace Transforms (Question and Solution Included).

[s3a]

I attached the question along with its solution.

Upon trying to find X(s), I get (s - 2)/(s^2 + 3s - 1) which is correct but after that I have to take the inverse Laplace transform and I don't know how to get the (-9/2 + sqrt(13)/2) and (-3/2 + sqrt(13)/2) parts and if someone could show me what the solution skipped, I would greatly appreciate it!

Thanks in advance!

 

[fluidistic]

I attached the question along with its solution.

Upon trying to find X(s), I get (s - 2)/(s^2 + 3s - 1) which is correct but after that I have to take the inverse Laplace transform and I don't know how to get the (-9/2 + sqrt(13)/2) and (-3/2 + sqrt(13)/2) parts and if someone could show me what the solution skipped, I would greatly appreciate it!

Thanks in advance!

Probably partial fractions from the roots of $s^2+3s-1$.

 

[vela]

I attached the question along with its solution.

Upon trying to find X(s), I get (s - 2)/(s^2 + 3s - 1) which is correct but after that I have to take the inverse Laplace transform and I don't know how to get the (-9/2 + sqrt(13)/2) and (-3/2 + sqrt(13)/2) parts and if someone could show me what the solution skipped, I would greatly appreciate it!

Thanks in advance!

Presumably you've inverted Laplace transforms before. Show us what you've tried.

 

[s3a]

I have done Laplace Transforms before and I think the partial fractions might be my problem.

For my first (logged) attempt, I tried a way without using partial fractions and got an answer but I may have made a mistake somewhere but I can't catch one. I anticipate a mistake since I was expected to use partial fractions and I get a seemingly different answer but I'm not sure if it's the same answer disguised in a different form.

For my second (logged) attempt, I used the quadratic formula since I was trying to use partial fractions and replicate the solution.

I'm attaching my work for what I mentioned above and would appreciate it if I can get the flaws of each attempt pointed out.

 

[vela]

In your first attempt, you made a sign error. When you completed the square, you should have gotten
$$s^2+3s-1 = s+3s+\left(\frac{3}{2}\right)^2 - \left(\frac{3}{2}\right)^2 - 1 = \left(s+\frac{3}{2}\right)^2 - \frac{13}{4}$$