When x is very small compared to the initial value, then we can reasonably assume in the calculation that terms such as "I �?x" or "I �?2x" are approximately equal to I, where "I" is an initial concentration. The 5% rule says that if the amount subtracted, such as x or 2x, is less than about 5% of the initial value, then using the approximation will not affect the calculated value of x if we round it to about two or three significant figures, which is typical.
In this problem, one of the equilibrium terms is 1.00 �?2x. Plugging the equilibrium terms into the equilibrium constant expression gives,
0.288 = (2x)2(x)
(1.00 �?2x)2
Approximating that 1.00 �?2x �?1.00, we have,
0.288 = (2x)2(x)
(1.00)2
Or, 0.288 = 4x3, x = 0.416 M.
Is this a valid answer? We'll test it with the 5% rule. To pass the 5% rule, 2x must be less than 5% of 1.00, since it was 2x, not just x, that was subtracted from the initial value of 1.00 M. Let's see:
2(0.416 M) X 100 = 83.2% 
1.00 M
Ouch! This percentage is much too high, so the approximation is invalid. Therefore, we have to calculate x much more accurately, which in this case, means solving a cubic equation for x without using the approximation. You will get three answers (roots), of which only one will either not be negative, or, will not give a negative concentration when inserted into 1.00 �?2x. All three solutions are valid mathematically, but physically, you cannot have negative concentrations.
Steve
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