Abstract We propose a two-parameter derivative-free family of methods with memory of convergence order 1.84 for finding the real roots of nonlinear equations. The new methods require only one function evaluation per iteration, so efficiency index is also 1.84. The process is carried out by approximating the derivative in Newton’s iteration using general quadratic equation αu2+βv2+α1u+β1v+δ=0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$alpha u^2+beta v^2+alpha _1 u+beta _1 v+delta =0$$end{document} in terms of coefficients α,βdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$alpha , beta $$end{document}. Various options of α,βdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$alpha , beta $$end{document} correspond to various quadratic forms viz. circle, ellipse, hyperbola and parabola. The application of new methods is validated on Kepler’s problem, Isentropic supersonic flow problem, L-C-R circuit problem and Population growth problem. In addition, a comparison of the performance of new methods with existing methods of same nature is also presented to check the consistency.
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