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Graphical solutions
Published in Surinder S. Virdi, Advanced Construction Mathematics, 2019
The general form of a cubic equation is ax3 + bx2 + cx + d = 0, where a, b, c and d are constants. The values of b, c, and d may be equal to 0, but constant a must have a value other than 0. A cubic equation can have 1, 2 or 3 real roots. If a cubic equation has 3 real roots, then their values will be different. However, there are cases where we could have only 1 or 2 solutions. A range of methods are available for solving a cubic equation, but here the graphical method will be used.
Numerical techniques
Published in C.W. Evans, Engineering Mathematics, 2019
We know how to solve quadratic equations; there is a simple formula for doing this. It is even possible to write a complicated set of procedures and formulas which will enable us to solve cubic equations and quartic equations explicitly. However, in general it is impossible to do this for polynomial equations of degree greater than four, and it is impossible to solve many other types of algebraic equation explicitly.
Multidimensional Effects, Potential Functions, and Fields
Published in Joel L. Plawsky, Transport Phenomena Fundamentals, 2020
and, in general, with this relation between a and s, we are left with cubic equations to solve for them. Cubic equations have either three real solutions or one real and two complex solutions. We are interested in the case where we have one real solution that we will denote by: a¯,s¯ Let's see what happens when we look at the stability of this particular solution. We assume a small perturbation about a¯,s¯ and substitute the perturbations into equations (9.168) and (9.169). For simplicity we will consider just a 1-D case and rewrite those equations in the following form: ∂a∂τ=Ra+β∂2a∂ξ2ξ=xL∂s∂τ=Rs+∂2s∂ξ2
A multi-dock, unit-load warehouse design
Published in IISE Transactions, 2019
Axiom 2.Expressing the expected dual-command distance as a function of the warehouse’s width and taking the first derivative with respect to the warehouse’s width, a cubic equation is obtained. For reasonable parameter values (the necessary condition for each scenario is provided in the proof of Corollary 4), the discriminant of the cubic equation is greater than zero. Therefore, the cubic equation has three distinct real roots, but there exist no rational roots, as the cubic equation is irreducible polynomial (from Galois Theory). Solving an irreducible cubic equation requires taking the roots of complex quantities. Therefore, reducing the cubic equation to a depressed form, setting the depressed cubic equation equal to zero, and solving for the warehouse’s width, the viable root can be obtained using Viète's trigonometric solution (Nickalls, 2006). The viable root is the first root, as results with the second and third roots are infeasible (the value of the expected distance is negative for the second root and the width of the warehouse is zero for the third root). Taking the second derivative of the expected dual-command distance with respect to the warehouse’s width and finding the second derivative is greater than zero for all reasonable values of the warehouse’s width establishes that the expected dual-command distance is a convex function of the warehouse’s width and the viable root is the optimal width of the warehouse.
Revisiting the Kellogg diagrams: roaster diagrams and their usefulness in pyrometallurgy
Published in Mineral Processing and Extractive Metallurgy Review, 2018
M. Sadegh Safarzadeh, Stanley M. Howard
The solution to this cubic equation may be obtained numerically using software such as EXCEL®, MATLAB®, or MathCad®. A function based on the differences in the left and right sides of the above equation as shown in Eq.(10) is used to find values at a fixed temperature such that Δ = 0. Imaginary roots and roots and are rejected.