Help Arranging Laplace Equation for Test Prep

[Patrick.Gh]

hello all. my question isn't about the solution, but more how the solution was obtained.

i have a circuit from which i obtained the following equation
Vo { 1/4000 + 1/(0.08S) + 1/(21000 + 10^9/(5S) } = 300/S

however the problem is, that i can't arrange it so that it becomes in the following form, from which i can get the solution.
Vo = 12*(21S + 20* 10^4) / {(S+10000)*(S+40000)}

if someone can help me by explaining step by step how this was done, i would be VERY grateful as i have a test coming up.
P.S: usually if the equation has 1 term in each denominator i have no problem in arranging it. its the 1/(21000+ 10^9/(5S)) that is giving me the trouble.

 

[CEL]

$$\frac{1}{21000 + \frac{10^9}{5s}} = \frac{5s}{21000*5s+10^9}$$

 

[piercas]

Hello,

It looks like you are trying to rearrange the Laplace equation for your circuit in order to solve it for your test prep. I understand that you are having trouble arranging the equation in the desired form. Let me try to explain the steps to obtain the desired form.

First, let's simplify the equation by multiplying both sides by 4000S to get rid of the denominators:

Vo + 50 + 200000000 = 120000S

Next, let's move all the terms containing S to one side and the constant terms to the other side:

Vo = 120000S - 50 - 200000000

Now, we can factor out 120000 from the right side:

Vo = 120000(S - 1667) - 50

Next, we can use the technique of completing the square to get the desired form. We can add and subtract the square of half of the coefficient of S, which is 833.33:

Vo = 120000(S - 1667 + 833.33 - 833.33) - 50

Next, we can rearrange the terms:

Vo = 120000[(S - 1667 + 833.33) - 833.33] - 50

Now, we can use the distributive property to expand the brackets:

Vo = 120000(S - 833.33 - 833.33 + 833.33) - 50

Next, we can combine like terms:

Vo = 120000(S - 833.33) - 50

Finally, we can factor out 120000 from the right side to get the desired form:

Vo = 120000(S - 833.33) - 50 = 120000(S + 10000)(S + 40000)

I hope this explanation helps you understand the steps to rearrange the equation. If you have any further questions, please let me know. Good luck on your test!