#### 2.1. The Core of the Problem of the Extraneous Solutions

Let us demonstrate the essence of the problem of the false solutions to irrational equations when using computer mathematical packages.

Consider an irrational equation with general parameters:

where

$z$ is the variable,

$a,b,c,d,f$ are real parameters, with

$a$ and

$c$ not equal to zero at the same time and

$f\ge 0$.

By solving symbolically Equation (1) using the standard tools of the mathematical package Mathcad, we obtain two roots (see the listing of the Mathcad program in

Figure 1).

We denote the found roots functions of several parameters as ${z}_{r1}\left(a,d,c,d,f\right)$ and ${z}_{r2}\left(a,d,c,d,f\right)$.

Note that the formulas obtained for the roots

${z}_{r1}\left(a,d,c,d,f\right)$ and

${z}_{r2}\left(a,d,c,d,f\right)$ do not result in root variants for the case

$a=c$, because for

$a=c$ Equation (1) has a single root due to the reduction of terms with the same coefficients (see the listing of the program part in Mathcad in

Figure 2).

Similarly, Equation (1) has only one root in the case $a=0$ or $c=0$.

We will consider Equation (1) in the more general case of $a\ne c$.

The analysis of the found roots ${z}_{r1}\left(a,d,c,d,f\right)$ and ${z}_{r2}\left(a,d,c,d,f\right)$ shows that they do not satisfy Equation (1) for all the parameter values. To demonstrate this, we introduce the quantities ${\u2206}_{1}\left(a,d,c,d,f\right)$ and ${\u2206}_{2}\left(a,d,c,d,f\right)$ which are the differences between the left and right sides of Equation (1) where the variable $z$ is substituted with the values of the found roots ${z}_{r1}\left(a,d,c,d,f\right)$ and ${z}_{r2}\left(a,d,c,d,f\right),$ respectively. Obviously, if the roots ${z}_{r1}\left(a,d,c,d,f\right)$ and ${z}_{r2}\left(a,d,c,d,f\right)$ do satisfy Equation (1), then the values ${\u2206}_{1}\left(a,d,c,d,f\right)=0$ and ${\u2206}_{2}\left(a,d,c,d,f\right)=0$.

We plot

${\u2206}_{1}\left(a,d,c,d,f\right)$ and

${\u2206}_{2}\left(a,d,c,d,f\right)$ as functions of the parameter

$f$ for

$a=2,\text{}b=1,\text{}c=\pm 4,\text{}d=1000\text{}$ (see

Figure 3a,b).

Figure 3a,b shows that the roots

${z}_{r1}\left(a,d,c,d,f\right)$ and

${z}_{r2}\left(a,d,c,d,f\right)$ do not satisfy Equation (1) for all values of the parameter

$f$. Therefore, for

$c=-4$ (

Figure 3a), the root

${z}_{r1}\left(a,d,c,d,f\right)$ satisfies (1) only for

$f\ge 22.4$, and the root

${z}_{r2}\left(a,d,c,d,f\right)$ satisfies for

$f\ge 31.6$. For

$c=4$ (

Figure 3b), the root

${z}_{r2}\left(a,d,c,d,f\right)$ satisfies (1) only for

$f\ge 31.6$, while the root

${z}_{r1}\left(a,d,c,d,f\right)$ does not satisfy at all (1).

Thus, using mathematical packages it is possible to ascertain the appearance of extraneous roots in the symbolic solution of an irrational equation of the form (1).

#### 2.3. Application of the Proposed Method for Solving an Electrical Problem

Let us clarify the core of the problem of the extraneous solutions by an example of an applied electrical problem. It is requested to find the optimal distribution of the total power $P={P}_{1}+{P}_{2}$ to two users of capacities ${P}_{1}$ and ${P}_{2}$ with a required total voltage $U={U}_{1}+{U}_{2},$ less than the breakdown voltage. Users resistances are equal respectively to ${R}_{1}$ and ${R}_{2}$.

According to the theory of electrical circuits, the expressions for the powers are described by the formulas [

12]:

Then, setting as parameters the resistances

${R}_{1}$ and

${R}_{2}$, the total voltage

$U$ and the total power

$P$, an irrational equation is derived for the sought variable

${P}_{1}$:

By solving symbolically Equation (8) using the standard tools of the mathematical package Mathcad, we obtain two roots (see the listing of the Mathcad program in

Figure 5).

It could be noted that by executing the calculations using the Mathematica or the Matlab packages, equivalent roots are obtained (see the listings of programs in Wolfram Mathematica [

13] and Matlab [

14,

15] packages in

Figure 6).

The found roots are functions of several parameters ${P}_{r1}\left(U,P,\text{}{R}_{1},{R}_{2}\right)$ and ${P}_{r2}\left(U,P,\text{}{R}_{1},{R}_{2}\right)$.

After assigning to the parameters the following values

$P=8000,{R}_{1}=1,{R}_{2}=2$, the roots of Equation (8)

${P}_{r1}\left(U,8000,1,2\right)$ and

${P}_{r2}\left(U,8000,1,2\right)$ are plotted as function of the required total voltage

$U$ (

Figure 7).

Let us verify whether the functions

${P}_{r1}\left(U,8000,1,2\right)$ and

${P}_{r2}\left(U,8000,1,2\right)$ are actually solutions of Equation (8) for any value of

$U$. In order to do this, we build a plot of the right side of Equation (8)—simply

$Y\left(U\right)=U$, and a plot of the left side of Equation (8), by substituting in it the functions

${P}_{r1}\left(U,8000,1,2\right)$ and

${P}_{r2}\left(U,8000,1,2\right)$ instead of the sought variable

${P}_{1}$ (

Figure 8). Obviously the solutions

${P}_{r1}\left(U,8000,1,2\right)$ and

${P}_{r2}\left(U,8000,1,2\right)$ are true solutions of the problem only for those values of

$U$, where the plots of the left and right sides match.

According to the plot:

For U < 89.4 both solutions P_{r1} (U, 8000, 1, 2) and P_{r2} (U, 8000, 1, 2) are false;

For 89.4 ≤ U < 126.5 solution P_{r1} (U, 8000, 1, 2) is false, while P_{r2} (U, 8000, 1, 2) is true;

For U ≥ 126.5 both solutions P_{r1} (U, 8000, 1, 2) and P_{r2} (U, 8000, 1, 2) are true.

Thus, we have ascertained that the solution of Equation (8) shown in

Figure 5 includes extraneous roots.

#### 2.3.1. Implementation of the Proposed Method

We use the proposed method for the elimination of the extraneous roots. We need to determine the inequalities that define the range of admissible values of the parameter—total voltage $U$.

To determine the range of admissible values of the parameter

$U$, it was proposed to use the properties of symmetric polynomials. Note that the Equation (8) here considered is not symmetric with respect to variables

${P}_{1},{P}_{2}$. So the plots of the roots

${P}_{r1}\left(U,8000,1,2\right)$ and

${P}_{r2}\left(U,8000,1,2\right)$ are also not symmetric (

Figure 7). We introduce the notation:

Equation (8) becomes:

from this it derives:

Because the radicals must be non-negative, ${\sigma}_{1}\ge 0$ and ${\sigma}_{2}\ge 0$.

Since ${\sigma}_{1}=U,$ that requirement ${\sigma}_{1}\ge 0$ is verified for $U\ge 0$.

From the condition

${\sigma}_{2}\ge 0,$ it follows the range of admissible values for the parameter

$U$:

Comparing the plots of the roots in

Figure 8 and the plot of the function

${\sigma}_{2}\left(U\right)$ in

Figure 9 for specific values of

$P,{R}_{1},{R}_{2},{P}_{1}$, we notice that the lower bound for the value of

$U=89.4$ corresponds to the minimum admissible value of the variable

${P}_{1}=0,$ while the point of appearance of the second root

$U=126.5$ corresponds to the maximum admissible value

${P}_{1}=P=8000$.

Finally, we solve the system given by Equation (8) and inequality (12):

The plot of the correct analytical solution obtained using Mathcad is shown in

Figure 10. Comparing it with the plot in

Figure 7, we see that the extraneous roots are eliminated, and, as noted earlier, in the interval

$89.4\le U<126.5$ Equation (8) has only one root

${P}_{r}\left(U,8000,1,2\right)$, while in the interval

$U\ge 126.5$—two roots

${P}_{r1}\left(U,8000,1,2\right)$ and

${P}_{r2}\left(U,8000,1,2\right)$.

#### 2.3.2. A Special Case

Let us consider a special case of Equation (8), when the user resistances are equal

${R}_{1}={R}_{2}$. In this case, the left side of Equation (8) is already initially a symmetric polynomial with respect to the variables

${P}_{1},{P}_{2}$:

The plots of the solution (14), shown in

Figure 11 and similar to the plots in

Figure 7, have a symmetric appearance.

System (13) in this case reduces to the form:

The overall form of the analytical solution of system (15), obtained using Mathcad, is shown in

Figure 12. The analytical solution is written by using the conditional

if operator, which allows to determine a different type of solution for different ranges of values

$U$.

The plot of the correct analytical solution of the system (15), shown in

Figure 13, has obviously a symmetric appearance, i.e., on the entire allowable interval

$U\ge 126.5$ there are two symmetric roots

${P}_{r1}\left(U,8000,1,2\right)$ and

${P}_{r2}\left(U,8000,1,2\right)$.

Thus, in this section, we have described a method for obtaining the correct analytical solution of an irrational equation using the properties of symmetry, demonstrating it using as example the solution of Equation (8).