Example 6 Given that the three numbers $a$, $b$, and $c$ satisfy the system of equations
$$
\left\{\begin{array}{l}
a+b=8, \\
a b-c^{2}+8 \sqrt{2} c=48 .
\end{array}\right.
$$

Try to find the roots of the equation $b x^{2}+c x-a=0$.
【Analysis】First, find the values of $ab$ and $a+b$ (expressed as algebraic expressions involving $c$), then construct a quadratic equation based on the relationship between roots and coefficients. Use the discriminant $\Delta \geqslant 0$ to determine the value of $c$, and subsequently find the values of $a$ and $b$.
Solution From the given conditions, we have
$$
a+b=8, \quad ab=c^{2}-8 \sqrt{2} c+48 \text{. }
$$

Then $a$ and $b$ are the two real roots of the equation
$$
x^{2}-8 x+c^{2}-8 \sqrt{2} c+48=0
$$

By the discriminant $\Delta \geqslant 0$, we get $c=4 \sqrt{2}$.
Thus, $a=b=4$.
Therefore, the original equation can be simplified to
$$
x^{2}+\sqrt{2} x-1=0 \text{. }
$$

Solving this, we get $x_{1}=\frac{-\sqrt{2}+\sqrt{6}}{2}, \quad x_{2}=\frac{-\sqrt{2}-\sqrt{6}}{2}$.