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When we use an ICE table, we often account for the missing information as the variable of X. We are usually given the chemical equilibrium constant, so we need to solve for X. We input all the chemical equilibrium concentrations/partial pressures into the chemical equilibrium expression even though concentrations/partial pressures may contain X or may only be represented by X. We try to equal this equation to 0 in order to solve for it through the quadratic equation. It will give you two values, but you always have to choose the one that is positive and/or smaller than the initial concentration/partial pressure that was provided. I hope this helps !
A lot of times when using ice tables, the equilibrium concentration equation will end up looking like x2/number-x = constant. Therefore, when you cross multiply by (number-x) and move it over to have one side of the equation equal to zero, you will end up with a quadratic equation, which can be solved by the quadratic equation.
When doing an ICE table, solving for x requires using a quadratic equation. However, if the K value given is less than 10^-3, then we can assume that the x value in the denominator is so close to zero that we can ignore it.
We use the quadratic equation in terms of chemical equilibrium when we are estimating the equilibrium point of a reaction through an ICE Table. If we are unable to use the "x is small" approximation, you must then use the quadratic equation to solve for x.
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