Example 2 On the blackboard, there are $n$ (where $n$ is an odd number) expressions: $* x^{2}+* x+*=0$. Two players, A and B, take turns to play the following game: each person is only allowed to replace one of the * symbols with a non-zero real number at a time. After a total of $3 n$ operations, $n$ quadratic equations are obtained. A tries to maximize the number of equations without real roots, while B tries to minimize the number of equations without real roots. Try to determine: Under the condition that A does not depend on how B operates, what is the maximum number of equations without real roots that A can achieve?
（19th All-Russian Mathematical Olympiad problem）
Jia each time changes the coefficient of $x$ in an untouched equation to 1, unless Yi changes the coefficient of $x^{2}$ or the constant term in an equation Jia has already changed to any real number $a \neq 0$. In this case, Jia immediately changes the constant term or the coefficient of $x^{2}$ in this equation to $\frac{1}{a}$, so the discriminant of this equation becomes $1-4 a \cdot \frac{1}{a} = 2 \sqrt{|c|}$. Thus, the discriminant of the resulting equation is greater than zero, meaning the equation has real roots. This way, Yi can change the coefficient of $x^{2}$ in at least $\frac{n-1}{2}$ untouched equations to 1, and ensure that these $\frac{n-1}{2}$ equations have real roots, thus leaving at most $n-\frac{n-1}{2}=\frac{n+1}{2}$ equations without real roots.
Therefore, the maximum value sought is $\frac{n+1}{2}$.