## 5 Determining the Roots of Interval Quadratic Equation  
By solving the classic crisp quadratic Equation (35) with exactly known coefficients \(a_{2},a_{1},a_{0}\) it is usually understood to find such values \(x_{1}\), \(x_{2}\) for which the left side of the equation takes the value 0, Figure 8.  
\[a_{2}x^{2}+a_{1}x+a_{0}=y=0 \tag{35}\]  
It should be noted that Equation (35) is a special case of the tolerant control problem (36). \[[\underline{a}_{2},\overline{a}_{2}]x^{2}+[\underline{a}_{1}, \overline{a}_{1}]x+[\underline{a}_{0},\overline{a}_{0}]=y,\quad y\in[ \underline{y}_{T},\overline{y}_{T}]=[0,0]\] \[\underline{a}_{2}=\overline{a}_{2},\ \ \underline{a}_{1}= \overline{a}_{1},\ \ \underline{a}_{0}=\overline{a}_{0},\ \ \underline{y}_{T}=\overline{y}_{T}=0 \tag{36}\]  
However, such an understanding of solving the quadratic equation is not always needed in practice. As an example, consider the fertilization equation for sugar beet (37). \[[-0.012,-0.010]x^{2}+[3.0,3.6]x+[250,450]=y,\quad y\in[0,0] \tag{37}\]  
Roots \(x_{1}\), \(x_{2}\) of Equation (37) denote the amount of nitrogen fertilizer [kg/ha] that should be applied to obtain a zero yield \(y\) [q/ha]. This information is unlikely to interest the farmer. He is interested in possible yield increases. However, there are other problems in which the equation roots are useful. Therefore, in the following, we will show how to determine the roots of the interval quadratic equation using the example of Equation (37), in a purely mathematical sense. Equation (37) can be interpreted as a tolerant control task in which the output \(y\) is required to be within the degenerate interval \(y\in[\underline{y}_{T},\overline{y}_{T}]=[0,0]\). Such a task can be solved mathematically using MIA. The solution is a set of pairs of roots conjugated by the RDM variables \(\gamma_{2},\gamma_{1},\gamma_{0}\in[0,1]\). However, none of the roots thus obtained has robustness to the uncertainty of the coefficients \((a_{2},a_{1},a_{0})\) of Equation (37). Under the conditions of uncertainty of these coefficients, the chance of hitting by the left side of  
Figure 8: Illustration of the sense of the roots \(x_{1}\), \(x_{2}\) of the quadratic equation \(a_{2}x^{2}+a_{1}x+a_{0}=y=0\) with exactly known values of the equation coefficients \(a_{2},a_{1},a_{0}\). Equation (37) into the tolerance interval \([\underline{y}_{T},\overline{y}_{T}]=[0,0]\) is infinitely small. In the case of the left side: \(L=[-0.012,-0.010]x^{2}+[3.0,3.6]x+[250,450]\), a realistic task is the task where the tolerance interval is not a degenerate zero but an extended zero [10]. Therefore, instead of (37), the problem (38) having a realistic tolerance range \([\underline{y}_{T},\overline{y}_{T}]=[-10,0,10]\) will be considered further. \[[-0.012,-0.010]x^{2}+[3.0,3.6]x+[250,450]=y,\quad y\in[-10,0,10] \tag{38}\]  
Formula (38) after transfer into a multidimensional form with RDM variables takes the form (39) and after further transformations the form (40). \[(-0.012+0.002\gamma_{a2})x^{2}+(3.0+0.6\gamma_{a1})x+(250+200\gamma_{a0})=-10 +20\gamma_{yT}\] \[\gamma_{a2},\gamma_{a1},\gamma_{a0},\gamma_{yT}\in[0,1] \tag{39}\]  
\[(-0.012+0.002\gamma_{a2})x^{2}+(3.0+0.6\gamma_{a1})x+250+200\gamma_{a0}+10- 20\gamma_{yT}=0\] \[a_{2}x^{2}+a_{1}x+a_{0}=y=0 \tag{35}\]0  
Note that the Equation (40) contains 5 variables--so it exists in the 5D space. The roots of Equation (40) can be calculated as:  
\[a_{2}x^{2}+a_{1}x+a_{0}=y=0 \tag{35}\]1 \[a_{2}x^{2}+a_{1}x+a_{0}=y=0 \tag{35}\]2 \[a_{2}x^{2}+a_{1}x+a_{0}=y=0 \tag{35}\]3  
Roots \(x_{1}(\gamma_{a2},\gamma_{a1},\gamma_{a0},\gamma_{yT})\) and \(x_{2}(\gamma_{a2},\gamma_{a1},\gamma_{a0},\gamma_{yT})\) are conjugated by the RDM variables. Only one pair of roots \((x_{1},x_{2})\) corresponds to one numerical combination of RDM variables \((\gamma_{a2},\gamma_{a1},\gamma_{a0},\gamma_{yT})\). All combinations of allowed values of RDM variables \((\gamma_{a2},\gamma_{a1},\gamma_{a0},\gamma_{yT})\) allow the generation of all possible pairs of roots.