Question

The parabola defined by y = (x − 3)² + 2 and the line defined by y = -x + 7 intersect at two points in the xy-coordinate plane. If (a, b) is the coordinate of one of the points of intersection and (c, d) is the other, what is the product of (a, b, c, and d)?

(a) 8
(b) 16
(c) 32
(d) 100

Answer 1

The parabola and line intersect at (3, 2) and (4, 3), and their product is 72.

Step-by-step explanation:

The parabola defined by the equation y = (x - 3)² + 2 and the line defined by the equation y = -x + 7 intersect at two points in the xy-coordinate plane. To find these points, we can set the two equations equal to each other:

(x - 3)² + 2 = -x + 7

Simplifying this equation, we get:

x² - 7x + 12 = 0

Using the quadratic formula, we can solve for x:

x = 3 or x = 4

Substituting these values of x back into either of the original equations, we can find the corresponding y-values:

For x = 3, y = (3 - 3)² + 2 = 2

For x = 4, y = (4 - 3)² + 2 = 3

Therefore, the two points of intersection are (3, 2) and (4, 3). The product of their coordinates is 3 * 2 * 4 * 3 = 72.