Abstract: This method is used for solving algebraic equation. This method reduce the problem to solving a second degree polynomial equation . This x-intercept will typically be a enhanced approximation to the function's root than the original guess, and the method can be iterated Based on collinear scaling and local quadratic approximation, quasi-Newton methods have improved for function value is not fully used in the Hessian matrix.one of the most important thing is that these method is not applicable for equation which has complex rule . this deficiency,obtaining a third order polynomial equation which has always real root . The Advantages of using Newton's method to approximate a root rest primarily in its rate of convergence. When the method converges, it does so quadratically. Also, the method is very simple to apply and has great local convergence. And the disadvantages of using this method are numerous. First of all, it is not guaranteed that Newton's method will converge if we select an {\displaystyle \displaystyle x_{0}}{\displaystyle \displaystyle x_{0}} that is too far from the exact root. Likewise, if our tangent line becomes parallel or almost parallel to the x-axis, we are not guaranteed convergence with the use of this method. Also, because we have two functions to evaluate with each iteration
({\displaystyle f(x_{k})}{\displaystyle f(x_{k})} and {\displaystyle f'(x_{k})}{\displaystyle f'(x_{k})}, this method is computationally expensive. Another disadvantage is that we must have a functional representation of the derivative of our function, which is not always possible if we working only from given data.

| DOI: 10.17148/IJIREEICE.2022.101202