Root, quadratic equation

This approximates the root x = 4.93488 from Program QROOT in only two steps. Solution by the quadratic equation yields x = 4.93487. 

Equation 5-247 is a polynomial, and the roots (C ) are determined using a numerical method such as the Newton-Raphson as illustrated in Appendix D. For second order kinetics, the positive sign (-r) of the quadratic Equation 5-245 is chosen. Otherwise, the other root would give a negative concentration, which is physically impossible. This would also be the case for the nth order kinetics in an isothermal reactor. Therefore, for the nth order reaction in an isothermal CFSTR, there is only one physically significant root (0 < C < C g) for a given residence time f. 

The roots can also be determined graphically (by using a graphing calculator, for instance) by noting where the graph of y(x) = ax1 + bx + c against x passes through y = 0 (Fig. 1). When a quadratic equation arises in connection with a chemical calculation, we accept only the... 

We quote below the results of computations for problem (3) with j/q = 1 and j/j =82, where is the smallest root to the quadratic equation (4). Once supplemented with those initial conditions, the exact solution of problem (3) takes the form j/, = i s (A = 0). Because of rounding errors, the first summand emerged in formula (5). This member increases along with increasing i, thus causing abnormal termination in computational procedures. 

If the transfer constants k, p and p are known, then the hybrid transfer constants a and P are the roots of the quadratic equation ... 

This quadratic equation has two roots P p and P p, which lead to two... 

Clearly, it would be best to avoid the need to have to make any assumptions about either the constancy of [G] or the relative magnitudes of [R T and [G]T. This can be done for the scheme of Eq. (1.38), and the outcome is a somewhat more complex expression for the concentration of AR G, which is obtainable from the roots of a quadratic equation ... 

Because a cannot exceed 1, only one root of this quadratic equation is acceptable and ... 

Cmax is the maximum concentration of solute that can be adsorbed by the soil, and K is the Langmuir adsorption coefficient. Combining the above two equations, a quadratic equation can be derived in terms of Cs, which on solving for the positive root gives... 

Mn(CO)5Br bears a close structural relationship to the hexacarbonyls so that the values of the ratios of its j>(CO)-derived polarizability tensors are of interest. For the axial carbonyl the ratio ai/a t is either —0.99 or —0.20 (the two roots of a quadratic equation obtained with neglect of L-matrix effects) compared with a value of ca. —0.23 for the hexacarbonyls08. The derived polarizability tensor for each... 

The quadratic equation always gives two answers. We can eliminate one of the answers since it is physically impossible to have a negative concentration. This leaves us with only the positive root. If we enter this answer into the bottom line of our table, we get the following equilibrium concentrations ... 

In the above equation ooF and ooG depend on the feed composition. They are the roots of the following quadratic equation, with coG>coF>0 ... 

In this problem, the right-hand side of the equilibrium expression is a perfect square. Noticing perfect squares, then taking the square root of both sides, makes solving the equation easier. It avoids solving a quadratic equation. 

Using the general solution for a quadratic equation, we can solve Eq. (6.67) for its two roots... 

The derivation of Equation (5.73) is dependent on the second law of thermodynamics and will be performed in Section 10.4.) Using Figure 5.8, we can see that Equation (5.73) (a quadratic equation in Tj) should have two distinct real roots for Tj at low pressures, two identical real roots at Pmax. and two imaginary roots above Pmax- At low pressure and high temperamre, which are conditions that correspond to the upper inversion temperature, the second term in Equation (5.73) can be neglected and the result is... 

The quadratic equation (10) has two roots and the shelf life is obtained by computing the root of Equation (10) that is smaller than a reference point, which is defined as... 

The reference point for this equation is xref = 41.89 and the initial point to accomplish convergence is xref - 7. The root for this equation is xR(l) = 33.25 and the shelf life for batch 1 is xL(l) = 33 months. Since the sampling times are the same for all batches, the following values remain invariant for all batches n = 12, x = 8, Sxx = 420, and to.o5,io = 1.8125. The values required by the quadratic equation and the associated shelf life for each batch are given in Table 27. The computer program to perform this calculation is given in the Appendix. 

The two roots of equation (3.15) may be abstracted by routine application of the quadratic formula. Before we do that, let us examine the results from three simplified special... 

The quartic expression for tr(J) conveniently only involves even powers of fi. The condition for the change of local stability, tr(J) = 0, therefore is a quadratic equation in fi2 with roots given by... 

The remaining possible solutions are given by the roots of a quadratic equation obtained by substituting from (12.47) into (12.50), yielding... 

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