Find the coordinates of the intersection points of the parabola y = x square -14 and the line x + y = 6

At the points of intersection of the graphs, the x and y coordinates of both functions are equal.

y = x² – 14; x + y = 6.

Express y from the second equation and substitute it into the first equation:

x + y = 6; y = 6 – x.

x² – 14 = 6 – x;

x² – 14 – 6 + x = 0;

x² + x – 20 = 0

We solve the quadratic equation using the discriminant:

a = 1; b = 1; c = -20;

D = b² – 4ac; D = 1² – 4 * 1 * (-20) = 1 + 80 = 81 (√D = 9);

x = (-b ± √D) / 2a;

x1 = (-1 – 9) / 2 = -10/2 = -5.

x2 = (-1 + 9) / 2 = 8/2 = 4.

Let’s calculate the value of y: y = 6 – x.

x = -5; y = 6 – 5 = 1. Point (-5; 1).

x = 4; y = 6 – 4 = 2. Point (4; 2).

Answer: The coordinates of the intersection points are (-5; 1) and (4; 2).