A parabola is given by the equation y = x^2 - 6 and an absolute-value curve is given by y = 4 - |x + 2| . If these two curves intersect at certain points, the sum of the y -coordinates of all valid points of intersection is

Step-by-step solution

To find the points of intersection, equate the two expressions for y : x^2 - 6 = 4 - |x + 2| x^2 + |x + 2| - 10 = 0 Case 1: x -2 The equation becomes x^2 + x + 2 - 10 = 0 x^2 + x - 8 = 0 The roots are x = -1 1 - 4(1)(-8) 2 = -1 33 2 Since 33 5.74 , the root x = -1 + 33 2 2.37 satisfies x -2 . The root x = -1 - 33 2 -3.37 does not satisfy x -2 and is rejected. For x = -1 + 33 2 , the y -coordinate is: y₁ = x^2 - 6 = (8 - x) - 6 = 2 - x = 2 - ( -1 + 33 2 ) = 5 - 33 2 Case 2: x The equation becomes x^2 - (x + 2) - 10