How do I solve for s?

[Nawaf]

How do I for solve for s in the equation [MATH] \frac{1}{s+a}[/MATH] [MATH]+[/MATH] [MATH]\frac{1}{s+b}[/MATH] [MATH] = [/MATH] [MATH]\frac{1}{c}[/MATH]

I started by multiplying both sides of the equation by c, so the final result was:

[MATH]\frac{c}{s+a}[/MATH] [MATH] + [/MATH] [MATH]\frac{c}{s+b}[/MATH] [MATH]= 1[/MATH]
Then I multiplied by the LCM which was (s+a)(s+b), so I got

[MATH]\frac{c(s+b)}{(s+a)(s+b)}[/MATH] [MATH]+[/MATH] [MATH]\frac{c(s+a)}{(s+b)(s+a)}[/MATH] [MATH]= 1[/MATH]
By adding the numerator of both fractions, I got

[MATH]\frac{c(s+b)+c(s+a)}{(s+a)(s+b)}[/MATH] [MATH]= 1[/MATH]
Distributive property for the numerator, I multiplied both terms in the denominator

[MATH]\frac{cs+cb+cs+ca}{s^2+sb+as+ab}[/MATH] [MATH]= 1[/MATH]
Cross multiply

[MATH]s^2+sb+as+ab = cs+cb+cs+ca[/MATH]
I tried to factor out c on the right side of the equation, but I always ended up with

[MATH]s^2+sb+a = c+cs+ca[/MATH]
I have no idea what I am supposed to do now, I am pretty sure that I need to solve for s using the quadratic equation but I don't know how to get rid of the right side of the equation and it is bothering me. This is my first time dealing with these types of equations.


Your help is very appreciated, thanks in advance!

 

[Cubist]

[MATH]s^2+sb+as+ab = cs+cb+cs+ca[/MATH]
I tried to factor out c on the right side of the equation, but I always ended up with

[MATH]s^2+sb+a = c+cs+ca[/MATH]

Hi, well done for showing us your work so far! I think that you made a mistake between the two lines I quoted above. If you can't work out what went wrong, then perhaps you can show us more detail about what you tried to do between those two lines?

FYI: I personally think that the top line is ready to be put into a quadratic form (I can't see any possible simplification myself). To do this I'd first move everything over to the LHS (subtract terms from both sides to do this). Then gather like terms in s, so that you end up with <something>*s^2 + <something>*s + <something>=0. None of the "<something>" expressions will contain the "s" variable.

 

[Steven G]

If you take the reciprocal of 1/c you would get c. So just compute the reciprocal of the other side and be done.

Multiplying both sides by c was what caused all your problems. Again just take the reciprocal of both sides