Problem 6. Find the value of $a$ for which the sum of all real roots of the equation

$$
\frac{f(a) \cdot x^{2}+1}{x^{2}+g(a)}=\sqrt{\frac{x g(a)-1}{f(a)-x}}
$$

is minimized, where $f(a)=a^{2}-\sqrt{21} a+26, g(a)=\frac{3}{2} a^{2}-\sqrt{21} a+27$. Provide the found value, rounding it to two decimal places if necessary.
Solution. Method 1. Note that for all values of $a: g(a)>f(a) \geqslant 20$, and therefore, the domain of the equation does not contain negative numbers. On the set $x \geqslant 0$, the functions $F(x)=\frac{f(a) \cdot x^{2}+1}{x^{2}+g(a)}$ and $F^{-1}(x)=\sqrt{\frac{x g(a)-1}{f(a)-x}}$ are mutually inverse. It is not difficult to prove that for each value of $a$ the function $F(x)=\frac{f(a) \cdot x^{2}+1}{x^{2}+g(a)}$ is monotonically increasing on the set $x \in[0 ;+\infty)$. Indeed, this follows from the representation $F(x)=f(a)+\frac{1-f(a) g(a)}{x^{2}+g(a)}$ and the inequality $1-f(a) g(a)0 \\
& h(2)=7-4 f(a)+2 g(a)=-a^{2}+2 \sqrt{21} a-430 \\
& g(a)=\frac{3}{2} a^{2}-\sqrt{21} a+27=\frac{3}{2}\left(a-\frac{\sqrt{21}}{3}\right)^{2}+\frac{47}{2} \geqslant \frac{47}{2}>0
\end{aligned}
$$

Let's factorize the first equation in the system. Therefore, the system is equivalent to the following:

$$
\left\{\begin{array}{l}
\left(x^{3}-x^{2} f+x g-1\right) \cdot\left(x^{2}\left(f^{2}+g\right)+x(f g-1)+\left(f+g^{2}\right)\right)=0 \\
x \neq f(a)
\end{array}\right.
$$

The quadratic equation $x^{2}\left(f^{2}+g\right)+x(f g-1)+\left(f+g^{2}\right)=0$ has no solutions, since

$$
D=(f g-1)^{2}-4\left(f+g^{2}\right)\left(g+f^{2}\right)=1-6 f g-2 f^{2} g^{2}-4 f^{3}-4 g^{3}0 \\
& h(2)=7-4 f(a)+2 g(a)=-a^{2}+2 \sqrt{21} a-430
\end{aligned}
$$

From this and the continuity of the function $h(x)$, it follows that for any value of $a$ there exist $x_{1} \in(0 ; 1)$, $x_{2} \in(1 ; 2), x_{3} \in(2 ; f(a))$ such that $h\left(x_{1,2,3}\right)=0$.

By Vieta's theorem, the sum of these three real roots is $f(a)=a^{2}-\sqrt{21} a+26$. The minimum of this expression is achieved at $a=\sqrt{21} / 2=2.291 \ldots \approx 2.29$

Answer: $a=\sqrt{21} / 2 \approx 2.29$.