Find the coordinates of the intersection points of the graphs of the functions y = x³

Find the coordinates of the intersection points of the graphs of the functions y = x³ in the numerator, x-2 in the denominator and y = x²-3x + 1

1. Let:

f (x) = x ^ 3 / (x – 2);
g (x) = x ^ 2 – 3x + 1.
2. Let us equate the functions to each other and find the coordinates of the points of intersection of their graphs:

x ^ 3 / (x – 2) = x ^ 2 – 3x + 1.

3. Multiply both sides of the equation by x – 2, taking into account that the value x = 2 cannot lead to a solution:

x ^ 3 = (x ^ 2 – 3x + 1) (x – 2).

4. Let’s open the brackets and solve the equation:

x ^ 3 = x ^ 3 – 3x ^ 2 + x – 2x ^ 2 + 6x – 2;
0 = -5x ^ 2 + 7x – 2;
5x ^ 2 – 7x + 2 = 0;
D = 7 ^ 2 – 4 * 5 * 2 = 9 = 3 ^ 2;
x1 = (7 – 3) / 10 = 4/10 = 0.4;
y1 = 0.4 ^ 2 – 3 * 0.4 + 1 = 0.16 – 1.2 + 1 = -0.04;
x2 = (7 + 3) / 10 = 10/10 = 1.
y2 = 1 ^ 2 – 3 * 1 + 1 = 1 – 3 + 1 = -1.
Answer: (0.4; -0.04), (1; -1).