83. Each quadratic equation can be reduced, by dividing it by the coefficient of the leading term, to the form

$$
x^{2}+c x+d=0
$$

The total number of different reduced quadratic equations in $p$-arithmetic is $p^{2}$. Calculate how many of them have no roots, how many have one root, and how many have two distinct roots.

1) The expression $b^{2}-4 a c$ is called the discriminant of the equation $a x^{2}+b x+c=0$.

## § 4. Extraction of a Cube Root. Simple Divisors of Numbers of the Form $a^{2}+3$
83. If $\alpha$ and $\beta$ are the roots of the equation $x^{2} + c x + d = 0$, then, as in ordinary algebra,

$$
x^{3} + c x + d = (x - \alpha)(x - \beta)
$$

(To convince yourself of this, it is sufficient to express $\alpha$ and $\beta$ in terms of $c$ and $d$ using the formulas for the solutions of a quadratic equation and expand the brackets.) Therefore, the number of reduced equations having two distinct roots is equal to the number of ways to choose two distinct numbers $\alpha$ and $\beta$ from the set of all $p$ numbers in $p$-arithmetic, i.e., it is equal to $\frac{p(p-1)}{2}$. If the equation $x^{2} + c x + d = 0$ has only one root $\alpha$, then

$$
x^{2} + c x + d = (x - \alpha)^{2}
$$

The number of such equations is $p$. All other equations, the number of which is $p^{2} - \frac{p(p-1)}{2} - p = \frac{p(p-1)}{2}$, have no solutions.