inverse laplace using partial fraction: How does 3x+11 equal 3 (s+3) + 2?

[johno149]

\(\displaystyle \mbox{Excercise:}\)

\(\displaystyle \mbox{Rewrite }\, \dfrac{3s\, +\, 11}{s^2\, +\, 6s\, +\, 13}:\, 3\, \dfrac{s\, +\, 3}{(s\, +\, 3)^2\, +\, 4}\, +\, \dfrac{2}{(s\, +\, 3)^2\, +\, 4}\)

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\(\displaystyle \mbox{Solution:}\)

\(\displaystyle \mbox{Rewrite }\, \dfrac{3s\, +\, 11}{s^2\, +\,6s\, +\,13}\, \mbox{ as }\, \dfrac{3s\, +\, 11}{(s\, +\, 3)^2\, +\, 4}\)

\(\displaystyle \mbox{Rewrite }\, \color{red}{ \dfrac{3s\, +\, 11}{(s\, +\, 3)^2\, +\, 4}\, \mbox{ as }\, \dfrac{3\, (s\, +\, 3)\, +\, 2}{(s\, +\, 3)^2\, +\, 4} }\)

\(\displaystyle \mbox{Rewrite }\, \dfrac{3\, (s\, +\, 3)\, +\, 2}{(s\, +\, 3)^2\, +\, 4}\, \mbox{ as }\, 3\, \dfrac{s\, +\, 3}{(s\, +\, 3)^2\, +\, 4}\, +\, \dfrac{2}{(s\, +\, 3)^2\, +\, 4}\)

\(\displaystyle 3\, \dfrac{s\, +\, 3}{(s\, +\, 3)^2\, +\, 4}\, +\, \dfrac{2}{(s\, +\, 3)^2\, +\, 4}\)

. . . . .\(\displaystyle =\, L^{-1}\, \left\{ 3\, \dfrac{s\, +\, 3}{(s\, +\, 3)^2\, +\, 4}\, +\, \dfrac{2}{(s\, +\, 3)^2\, +\, 4} \right\}\)

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Hi, I am new to this forum so was just wondering if anyone can help on this issue. I am trying to understand how the numerator (in the red portion above) changes from 3s+11 to 3(s+3)+2 in the worked example given in the picture attached. Any help greatly appreciated

 

[Deleted member 4993]

\(\displaystyle \mbox{Excercise:}\)

\(\displaystyle \mbox{Rewrite }\, \dfrac{3s\, +\, 11}{s^2\, +\, 6s\, +\, 13}:\, 3\, \dfrac{s\, +\, 3}{(s\, +\, 3)^2\, +\, 4}\, +\, \dfrac{2}{(s\, +\, 3)^2\, +\, 4}\)

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\(\displaystyle \mbox{Solution:}\)

\(\displaystyle \mbox{Rewrite }\, \dfrac{3s\, +\, 11}{s^2\, +\,6s\, +\,13}\, \mbox{ as }\, \dfrac{3s\, +\, 11}{(s\, +\, 3)^2\, +\, 4}\)

\(\displaystyle \mbox{Rewrite }\, \color{red}{ \dfrac{3s\, +\, 11}{(s\, +\, 3)^2\, +\, 4}\, \mbox{ as }\, \dfrac{3\, (s\, +\, 3)\, +\, 2}{(s\, +\, 3)^2\, +\, 4} }\)

\(\displaystyle \mbox{Rewrite }\, \dfrac{3\, (s\, +\, 3)\, +\, 2}{(s\, +\, 3)^2\, +\, 4}\, \mbox{ as }\, 3\, \dfrac{s\, +\, 3}{(s\, +\, 3)^2\, +\, 4}\, +\, \dfrac{2}{(s\, +\, 3)^2\, +\, 4}\)

\(\displaystyle 3\, \dfrac{s\, +\, 3}{(s\, +\, 3)^2\, +\, 4}\, +\, \dfrac{2}{(s\, +\, 3)^2\, +\, 4}\)

. . . . .\(\displaystyle =\, L^{-1}\, \left\{ 3\, \dfrac{s\, +\, 3}{(s\, +\, 3)^2\, +\, 4}\, +\, \dfrac{2}{(s\, +\, 3)^2\, +\, 4} \right\}\)

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Hi, I am new to this forum so was just wondering if anyone can help on this issue. I am trying to understand how the numerator (in the red portion above) changes from 3s+11 to 3(s+3)+2 in the worked example given in the picture attached. Any help greatly appreciated :smile:

Are you working with paper/pencil or just staring at the screen??!!

3(s+3) + 2 = (3s + 9) + 2 = 3s + 9 + 2 = 3s + 11 ← super-simple arithmetic

 

[johno149]

Are you working with paper/pencil or just staring at the screen??!!

3(s+3) + 2 = (3s + 9) + 2 = 3s + 9 + 2 = 3s + 11 ← super-simple arithmetic

I'll admit there was a bit of staring Thank you very much