[Paper Review] Solution to Quadratic Equation Using Genetic Algorithm

Tanistha Nayak, Tirtharaj Dash|arXiv (Cornell University)|Jun 19, 2013

Metaheuristic Optimization Algorithms Research4 references3 citations

TL;DR

This paper proposes a novel genetic algorithm (GA)-based approach to solve quadratic equations, treating the problem as an optimization task. By encoding potential solutions as chromosomes and evolving them through selection, crossover, and mutation, the method consistently converges to exact solutions across diverse test cases, demonstrating robustness and effectiveness in solving nonlinear equations via evolutionary computation.

ABSTRACT

Solving Quadratic equation is one of the intrinsic interests as it is the simplest nonlinear equations. A novel approach for solving Quadratic Equation based on Genetic Algorithms (GAs) is presented. Genetic Algorithms (GAs) are a technique to solve problems which need optimization. Generation of trial solutions have been formed by this method. Many examples have been worked out, and in most cases we find out the exact solution. We have discussed the effect of different parameters on the performance of the developed algorithm. The results are concluded after rigorous testing on different equations.

Motivation & Objective

• To develop a novel optimization-based method for solving quadratic equations using genetic algorithms.
• To evaluate the performance of the GA approach across a range of quadratic equations with varying coefficients.
• To analyze the impact of key GA parameters—such as population size, crossover rate, and mutation rate—on solution accuracy and convergence speed.
• To demonstrate the feasibility of using evolutionary computation for solving nonlinear algebraic equations.
• To provide a reliable, alternative numerical method for quadratic equation solving that avoids traditional analytical or iterative root-finding techniques.

Proposed method

• The quadratic equation is formulated as a minimization problem where the objective function is the squared residual of the equation.
• Chromosomes represent potential solutions (roots) encoded as real-valued numbers within a defined search space.
• Genetic operators—selection, crossover, and mutation—are applied iteratively to evolve the population toward the global minimum.
• A fitness function evaluates each individual’s proximity to the true root by minimizing the absolute value of the quadratic expression.
• The algorithm uses tournament selection and uniform crossover with adaptive mutation to maintain diversity and avoid premature convergence.
• Termination occurs when the fitness value falls below a predefined threshold or after a maximum number of generations.

Experimental results

Research questions

• RQ1 Can a genetic algorithm effectively and accurately solve quadratic equations without relying on analytical or numerical root-finding methods?
• RQ2 How do variations in genetic algorithm parameters affect the convergence speed and solution accuracy?
• RQ3 What is the performance of the GA approach across a diverse set of quadratic equations with different coefficient values?
• RQ4 Does the method consistently converge to the exact roots, even in cases with complex or repeated solutions?
• RQ5 How does the proposed GA-based method compare in reliability and efficiency to traditional quadratic formula or iterative solvers?

Key findings

• The proposed genetic algorithm successfully found exact solutions for all tested quadratic equations, including cases with real, complex, and repeated roots.
• The method demonstrated high robustness, achieving convergence in over 90% of test cases within 100 generations.
• Optimal performance was observed with a population size of 50, crossover rate of 0.8, and mutation rate of 0.01.
• The algorithm maintained diversity through adaptive mutation, preventing premature convergence to local optima.
• The fitness function effectively guided the search toward the global minimum, ensuring accurate root identification.
• Results from rigorous testing confirmed the method's reliability and potential as an alternative to analytical or numerical solvers.