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Yes, because the change to the initial concentration of the acid or base will be essentially zero. But usually the percent ionization doesn't come first, and is a step to confirm that approximating was okay (which you would do based on the value of K being <10^-3).
Yes as long as the percent ionization is less than 5% of the initial concentration then the approximation is okay. If the equilibrium constant is less than 10^-3 then you do not have to use the quadratic equation and to check, you would calculate the percent ionization as confirmation.
If you don't use the quadratic equation, you still want to justify your reason for ignoring the "-x" terms in your equilibrium by saying that since x is less than 5% of the initial concentration, its change is negligible.
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