Find the intersection points of the line y = -2x-3 with the parabola y = x2 + 4x-10

Since at the point of intersection of the straight line y = -2x – 3 with the parabola y = x ^ 2 + 4x – 10, they will have common coordinates, we equate the right-hand sides of these functions:

x ^ 2 + 4x – 10 = -2x – 3.

Move everything to the left side and give similar terms:

x ^ 2 + 4x – 10 + 2x + 3 = 0;

x ^ 2 + 6x – 7 = 0.

To solve this quadratic equation, we write down its coefficients and calculate the discriminant:

a = 1; b = 6; c = -7;

D = b ^ 2 – 4ac = 6 ^ 2 – 4 * 1 * (-7) = 64 – so the equation has two solutions, and the parabola and the straight line have two intersection points.

x1 = (-b + √D) / (2a) = (-6 + √64) / (2 * 1) = 1;

x2 = (-b – √D) / (2a) = (-6 – √64) / (2 * 1) = -7.

We are looking for coordinates at:

y1 = -2 * 1 – 3 = -2 – 3 = -5;

y1 = -2 * (-7) – 3 = 14 – 3 = 11.

Answer: (1; -5), (-7; 11).