For which value of the parameter $a$ will the two roots of the quadratic equation $x^{2}-4 a x+5 a^{2}-6 a=0$ be the farthest apart?
Solution. For the quadratic equation $a x^{2}+b x+c=0$ to have two distinct roots, the discriminant must be greater than zero. The larger the discriminant, the larger the number we need to subtract from or add to $-\frac{b}{2 a}$. The more we subtract or add, the greater the difference between the two roots. The value of the discriminant (denoted as $D$) in the given equation is:

$$
D=f(a)=16 a^{2}-4\left(5 a^{2}-6 a\right)=-4 a^{2}+24 a=4 a(6-a) .
$$

The equation $f(a)=0$ has two roots $a_{1}=0$ and $a_{2}=6$. Since the coefficient of $a^{2}$ is $-4<0$, the graph of the function $f(a)$ is a parabola opening "downwards," with its maximum at the average of the two zeros. Therefore, $D$ is the largest when $a=3$, and at this point, the roots of the original equation will be the farthest apart.