Question

Tom determines that the system of equations below has two solutions, one of which is located at the vertex of the parabola.
Equation 1: (x – 3)2 = y – 4
Equation 2: y = -x + b
In order for Tom’s thinking to be correct, which qualifications must be met?
b must equal 7 and a second solution to the system must be located at the point (2, 5).
b must equal 1 and a second solution to the system must be located at the point (4, 5).
b must equal 7 and a second solution to the system must be located at the point (1, 8).
b must equal 1 and a second solution to the system must be located at the point (3, 4).

Answer 1

The parabola is y = (x − 3)^2 + 4, whose vertex is (3,4). For (3,4) to lie on y = −x + b, plug in: 4 = −3 + b, so b = 7.
With b = 7, solve intersections: (x − 3)^2 + 4 = −x + 7 ⇒ (x − 3)^2 + x − 3 = 0 ⇒ x^2 − 5x + 6 = 0 ⇒ (x − 2)(x − 3) = 0.
Thus the solutions are x = 3 (vertex) and x = 2, giving the second solution (2,5).

So: b = 7 and the other solution is (2,5).