Parabola a can be represented using the equation (x + 3)2 = y, while line b can be represented using the equation y = mx + 9. isabel determines one solution to the system of two equations must always pass through the vertex of parabola a. which best describes the reasonableness of her solution?

Parabola a:   y = (x+3)²
line b: y = mx + 9

y = (x+3)²  → (x+3)(x+3) = x² + 3x + 3x + 9 = x² + 6x + 9

y = y
mx + 9 = x² + 6x + 9 ; assume that m = 1
x + 9 = x² + 6x + 9
9 = x² + 6x - x + 9
9 - 9 = x² + 5x
0 = x² + 5x

x = -b +  √(b²) - 4ac / 2a
x = [-5 +  √25 - 4(1)(0)] /2(1)
x = (- 5 +  √25)/2 = -5 + 5 / 2 = 0/2 = 0
x = (-5 -  √25)/2 = -5 -5 /2 = -10/2 = -5

x = 0 ; y =  x² + 6x + 9  → y = 0² + 6(0) + 9  → y = 9   → (0,9)
x = -5 ; y = x² + 6x + 9  → y = (-5)² + 6(-5) + 9  → y = 25 - 30 + 9 → y   = 4   → (-5,4)

the points of intersection are (0,9) and (-5,4)