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!
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!