Weakly Basic Solution OH- Concentration

[Big-Daddy]

Hi, sorry if this is a stupid mistake or I should be ignoring Wikipedia, I'd just like clarity.

 

As I understand it, the concentration of H3O+ ions in an acidic solution is denoted by (x^3)+(Ka*(x^2))-((Kw*x)+(C*Ka*x))-(Kw*Ka)=0 (where x is the concentration of H3O+ ions, C is the starting concentration of acid molecules, Ka is the acid dissociation constant and Kw is the auto-ionization constant of water.

 

My question is, assuming that the above is correct (if it's not please correct me), is it safe to describe the concentration of OH- ions in an alkaline solution with (x^3)+(Kb*(x^2))-((Kw*x)+(C*Kb*x))-(Kw*Kb)=0 (where x is now the concentration of OH- ions instead)?

 

The doubt arises because Wikipedia says that "For alkaline solutions an additional term is added to the mass-balance equation for hydrogen. Since addition of hydroxide reduces the hydrogen ion concentration, and the hydroxide ion concentration is constrained by the self-ionization equilibrium to be equal to Kw/[H+]", so it appears that, in the alkaline equation, I should simply keep x=concentration of H3O+ and replace all the current 'x's with 'Kw/x' (which will thus be equal to the concentration of OH-). But I don't see why that's necessary - can't I just use (x^3)+(Kb*(x^2))-((Kw*x)+(C*Kb*x))-(Kw*Kb)=0 where x is the concentration of OH- ions for my calculation? If I want to calculate H3O+ ions additionally, can't I just use [H3O+]*[OH-]=Kw after I solve the cubic above?

 

E.G. If I want to calculate the H3O+ ionic concentration in an alkaline solution, can't I just use (x^3)+(Kb*(x^2))-((Kw*x)+(C*Kb*x))-(Kw*Kb)=0 where x is the concentration of OH- ions and then use [H3O+]=Kw/[OH-]? If so, why does Wikipedia suggest using a Kw/[H3O+] system? If not, where am I going wrong?

 

Thanks a lot for any help.