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The quadratic formula will naturally end up being used in ICE tables typically when the unknown concentrations for the products in equilibrium are the same and the reactant at equ. has the initial conc. minus the change in conc. This is because, when the K equation is written out with these values, it will look something like (x^2) / (a-x), and when manipulated algebraically to find x, the quadratic equation will have to be used.
I've been told that we can use the quadratic formula when the value is greater than 10^-3. You can also use the 5% rule to double check if the value is eligible to make x equal to zero and to completely ignore it since it's infinitesimal.
You can use the 5% approximation when K<10^-3 (when the reaction strongly favors the reactants). This means that no quadratic equation solving will be necessary. But if this is not the case and the reaction doesn't strongly favor the reactants, you will end up having to solve a quadratic.
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