Tom determines 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 this solution to be reasonable, which qualifications must be met?

a) b must equal 7 and a second solution to the system must be located at the point (2, 5).
b) b must equal 1 and a second solution to the system must be located at the point (4, 5).
c) b must equal 7 and a second solution to the system must be located at the point (1, 8).
d) b must equal 1 and a second solution to the system must be located at the point (3, 4).

Option A is the correct answer.

Step-by-step explanation:

In this question the given parabola is (x - 3)²= y - 4

And the equation is y = -x + b

One solution of these equations lies on vertex.

Therefore we have to find the coordinates of vertex so that value of b can be measured.

We know that if a parabola is in the form of y = (x - h)²+ k

Then the coordinates of the vertex of this parabola will be (h, k).

Now our equation of parabola is y = (x-3)² + 4

Therefore the vertex will be (3, 4).

And it has been given that the vertex is y = -x + b

Therefore by putting x = 3 and y = 4 we can get the value of b.

4 = -3 + b

b = 7.

Now the second solution to the system will be

(x - 3)² = -(x - 7) - 4

x² + 9 - 6x = -x + 7 - 4

x² +9 - 6x = -x + 3

x² - 6x + x + 9 - 3 = 0

x² -5x + 6 = 0

x² - 3x - 2x + 6 = 0

x(x - 3) - 2(x - 3) = 0

(x - 3)(x -2 ) = 0

x = 3, 2

Hence y = (-3 + 7) and y = (-2 + 7)

y = 4, 5

As we know first solution is (3, 4) the second solution will be (2, 5).

Option A is the correct answer.

part a: 1.08

part b: 27.59