Find the abscissas of the intersection points of the graphs of the functions y = x ^ 2 and y = 6-x.

To find the abscissas of the intersection points of the graphs of the function, given by the equations y = x ^ 2 and y = 6 – x, we compose and solve a quadratic equation.

At the intersection points, the ordinates of the functions are the same, so we can write the equation:

x ^ 2 = 6 – x;

x ^ 2 + x – 6 = 0;

Solve the full quadratic equation using the discriminant.

D = b ^ 2 – 4ac = 1 ^ 2 – 4 * 1 * (- 6) = 1 + 24 = 25;

x1 = (- b + √D) / 2a = (1 + √25) / 2 * 1 = (- 1 + 5) / 2 = 4/2 = 2;

x2 = (- b – √D) / 2a = (1 – √25) / 2 * 1 = (- 1 – 5) / 2 = – 6/2 = – 3.

Answer: abscissas of the intersection points x1 = 2; x2 = – 3.