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Engineering and Scientific Calculations
Published in David E. Clough, Steven C. Chapra, Introduction to Engineering and Scientific Computing with Python, 2023
David E. Clough, Steven C. Chapra
In evaluating the quadratic formula, the term b2–4ac is called the discriminant because, depending on its sign, very different solutions arise. When the discriminant is positive, two distinct real roots result. When it is zero, there are two equal real roots. When b2–4ac < 0, we are faced with the awkward situation of finding the square root of a negative number. This is handled by defining a symbol, j, to represent the square root of −1 as j≜−1
Solving quadratic equations
Published in John Bird, Bird's Engineering Mathematics, 2021
If the quadratic expression can be factorised this provides the simplest method of solving a quadratic equation. For example, if2x2−5x−3=0,then,by factorising:(2x+1)(x−3)=0Hence either(2x+1)=0i.e.x=−12or(x−3)=0i.e.x=3
Errors in algebra
Published in Breach Mark, Essential Maths for Engineering and Construction, 2017
We know that a pair of independent simultaneous equations in two variables has one solution for each of the variables. We also know that a quadratic equation can have up to two distinct real solutions. Therefore, will a pair of simultaneous quadratic equations in x and y have up to two distinct solutions for the pair (x,y)?
Estimating the boundary of the region of attraction of Lotka–Volterra system with time delays
Published in Systems Science & Control Engineering, 2021
Juanjuan Yang, Rui Dong, Juntao Wang
Two-species competitive L-V type system is a quadratic system. Quadratic systems play an important role in the modelling of a wide class of nonlinear processes (electrical, robotic, biological, etc.). For such systems, it is of mandatory importance not only to determine whether the origin of the state space is locally asymptotically stable but also to ensure that the operative range is included into the convergence region of the equilibrium. Over the years, several papers have focused on the estimate of the RA of the zero-equilibrium point of quadratic system (Caldeira et al., 2018; Chen et al., 2013; Chesi et al., 2005; Chiang et al., 1988; Genesio & Vicino, 1984; Li-Ya et al., 2019; Merola et al., 2017; Tesi et al., 1996; Xu et al., 2020). In Genesio and Vicino (1984), a Lyapunov-based procedure is proposed to compute an ellipsoidal estimate of the RA of a quadratic system. Considering this method is computationally heavy, an estimate of the RA based on topological considerations was provided in Chiang et al. (1988) or based on linear matrix inequalities (LMIs) feasibility problem (Chesi et al., 2005; Tesi et al., 1996).