## 1 Introduction  
Scientists, engineers, and practitioners have been trying to apply classical mathematics to solving real problems for centuries. However, this mathematics requires precise knowledge of both the formulas describing the dependencies existing in the system and the exact values of the coefficients occurring in them. Unfortunately, in real systems, we often do not fully know either one or the other. For this reason, classical mathematics is sometimes not suitable for solving real problems, which makes scientists try to develop new mathematical methods capable of analyzing and processing uncertain, approximate data, e.g., interval, fuzzy data, etc. An example of the problem with uncertainty is presented in [1]. It concerns vendor selection and supply chain management. The problem describes a complex delivery system whose future behavior cannot be accurately predicted. Taking into account the uncertainty of this system, it is true that only uncertain results can be obtained. But the results are realistic and do not introduce the illusion of accuracy, which in this case is unattainable. The second example of an uncertain problem is the task of making optimal decisions presented in [2]. Decision optimization in the conditions of having uncertain data (here fuzzy) requires the use of more sophisticated mathematical methods than in the case of precise data. Various mathematical forms of defining uncertain data are used in science. However, the most commonly used form is the interval. Confirmation of this truth is provided by articles such as [3,4] recently published in Applied Sciences. In [3] the authors apply interval analysis to the complicated problem of heat and mass transfer. It is a problem where it is difficult to find exact data values. Another very practical example of using intervals is the behavior analysis of a two-pole generator rotor systems [4]. The authors of this article have been dealing with solving uncertainty problems with particular emphasis on equations for many years. Solving uncertain equations is a much more difficult task than calculating the values of uncertain functions. In an article [5] published in 2022 in Applied Sciences, the authors presented a way to realistically solve the basic interval equation \([\underline{a},\overline{a}]x=[y_{\tau},\overline{y}_{\tau}]\) based on the newly introduced concept of control value robustness to a set of possible, uncertain system states. The task of solving interval and fuzzy quadratic equations is very complicated and has been dealt with by many scientists. It should be remembered here that an interval is a special case of a fuzzy set whose membership function is rectangular. Hence, the methods of solving fuzzy equations can be used to solve interval equations. Further on, the solution of the quadratic interval equation (QIE) in the form (1) will be discussed. \[[\underline{a}_{2},\overline{a}_{2}]x^{2}+[\underline{a}_{1},\overline{a}_{1 }]x+[\underline{a}_{0},\overline{a}_{0}]=[\underline{y}_{\tau},\overline{y}_ {\tau}] \tag{1}\]  
Equation of type (1) will be applied further in solving the problem of fertilizing plants. Solving interval equations (IE) and fuzzy equations (FE) is a very difficult task, as the history of their research has shown. Already [PERSON] wrote about it in 1964 and 1965 [6,7] and [PERSON] in 1982 [8]. They emphasized that one of the reasons may be the multidimensional nature of the solution (\(X_{1}\times X_{2}\) space) and the complicated and non-convex shape of the set of possible solutions. In recent years it was also pointed out by [PERSON] and [PERSON] in [9] and [PERSON] and [PERSON] in [10,11], who state that in the case of nonlinear, quadratic interval equations \"to date there are no universal methods for solving such equations and that the problem is open\". The difficulties with solving linear and nonlinear IEs and FEs are also well presented in the articles of [PERSON] [12,13,14,15].