Extraneous roots

Math problems sometimes require solving algebraic equations. The algebra is wonderful, but you run the risk of creating some extraneous roots. Extraneous roots or solutions are answers that satisfy the algebraic equation but don t really mean a thing in the situation. 

While the iterative scheme adopted did not, in this instance, lead to extraneous roots, this possibility can be avoided completely by using the form given in the Appendix (Equation 23) for any future least squares estimation. This form does not possess the unwanted root. 

Solving a quadratic equation always yields two roots. One root (the answer) has physical meaning. The other root, while mathematically correct, is extraneous that is, it has no physical meaning. The value of x is defined as the number of moles of A per liter that react and the number of moles of B per liter that react. No more B can be consumed than was initially present (0.100 M), so x = 0.309 is the extraneous root. Thus, x = 0.099 is the root that has physical meaning, and the extraneous root is 0.309. The equilibrium concentrations are... 

This formula gives two roots, both of which are mathematically correct. A foolproof way to determine which root of the equation has physical meaning is to substimte the value of the variable into the expressions for the equilibrium concentrations. For the extraneous root, one or more of these substimtions will lead to a negative concentration, which is physically impossible (there cannot be less than none of a substance present ). The correct root will give all positive concentrations. In Example 17-7, substitution of the extraneous root x = 0.309 would give [A] = (0.300 — 0.309) M = —0.009 M and [B] = (0.100 — 0.309) M = —0.209 M. Either of these concentration values is impossible, so we would know that 0.309 is an extraneous root. bu should apply this check to subsequent calculations that involve solving a quadratic equation. 

Clearly, x = —0.25 is the extraneous root, because x cannot be less than zero in this case. This reaction does not consume a negative quantity of HI, because the reaction is proceeding toward the left. Thus, x = 0.032 is the root with physical meaning, so the new equilibrium concentrations are... 

Note that m, yand y2 are known quantities that are calculated from the initial concentrations a and fc, and the equilibrium constant K. The roots, y and y2, are the possible equilibrium values of y y is an extraneous root and y2 is the equilibrium value. 

The prediction trend value develops with extraneous root of Yf The closer it is to the forecast period, the closer the trend values and forecast values will be, which means a better effect of the prediction model ... 

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