Calculate the coordinates of the intersection of the parabola y = x ^ 2 + 3x – 1 and the hyperbola y = 3 / x. I get 3 points.

Consider two functions y = x² + 3 * x – 1 and y = 3 / x. As is known, the first function on the coordinate plane is represented by a parabola, and the second function is represented by a hyperbola. In the task, you need to calculate the coordinates of the intersection points of the parabola and hyperbola. In order, first, to fulfill the requirement of the task, equating the right-hand sides of these equalities, we get: x² + 3 * x – 1 = 3 / x. We multiply both sides of the resulting equality by x, and then, collect all the terms on the left side of the equality x³ + 3 * x² – x – 3 = 0. Solve the resulting cubic equation.
Let’s use the so-called distribution property of multiplication with respect to addition (subtraction), which in formal notation has the form: a * (b ± c) = a * b ± a * c. It should be noted that the above property can be applied in the reverse order. We have: x² * (x + 3) – (x + 3) = 0 or (x + 3) * (x² – 1) = 0.
For the product of two factors to be equal to zero, a necessary and sufficient condition is the equality of at least one of the factors to zero. Using this fact, we get: x = -3; x = -1 and x = 1. These roots of the last equation are the abscissas of the intersection points of the parabola and hyperbola. Let us calculate the corresponding ordinates of the intersection points. When x = -3, we get the ordinate y = 3 / (-3) = -1; similarly, for x = -1, we have y = 3 / (-1) = -3; similarly, if x = 1, then y = 3/1 = 3. The found points will be denoted by A (-3; -1), B (-1; -3) and C (1; 3).
Answer: (-3; -1), (-1; -3) and (1; 3).