Without performing construction, find the coordinates of the points of intersection of the parabola .

Without performing construction, find the coordinates of the points of intersection of the parabola y = x ^ 2-14 and the straight line x + y = 6.

To find the coordinates of the intersection points of the graphs of the functions given by the formulas y = x ^ 2 – 14 and x + y = 6, it is necessary to combine these equations into a system and find its solutions.

y = x ^ 2 – 14; x + y = 6 – we express from the second equation of the system y through x;

y = x ^ 2 – 14; y = 6 – x – equate the right sides of these equations;

x ^ 2 – 14 = 6 – x;

x ^ 2 – 14 – 6 + x = 0;

x ^ 2 + x – 20 = 0;

D = b ^ 2 – 4ac;

D = 1 ^ 2 – 4 * 1 * (- 20) = 1 + 80 = 81; √D = 9;

x = (- b ± √D) / (2a);

x1 = (- 1 + 9) / 2 = 8/2 = 4;

x2 = (- 1 – 9) / 2 = – 10/2 = – 5.

To find the corresponding values ​​of y, it is necessary to substitute the found values ​​of x into any of the equations of the system, for example, into the second y = 6 – x.

y1 = 6 – x1 = 6 – 4 = 2;

y2 = 6 – x2 = 6 – (- 5) = 6 + 11 = 17.

Answer. (4; 2); (- 5; 17).