The points (3, 8) and (6, 8) are part of the solution set of a quadratic inequality, while (5, 9) and (7, 7) are not. The point that must also be part of the solution set is
A. (4, 7)
B. (4, 9)
C. (8, 6)
D. (6, 10)
(3,8) and (6,8) are part of the solution set, the vertex is at x = 9/2, so
y = a(x - 9/2)^2 + b
so we know that
a(3 - 9/2)^2 + b = 8
a(6 - 9/2)^2 + b = 8
or, 9a+4b = 32
The problem with that is that there are many values of a which fit those two points. A parabola is determined by 3 points. So you could pick an a such that none or all of the choices lie in the solution set.
To determine which point must also be part of the solution set, we need to understand the characteristics of quadratic inequalities.
A quadratic inequality is an inequality that includes a quadratic expression. It can be in the form of Ax^2 + Bx + C < 0, Ax^2 + Bx + C > 0, Ax^2 + Bx + C ≤ 0, or Ax^2 + Bx + C ≥ 0.
To find the equation of the quadratic inequality, we can use the given points (3, 8) and (6, 8). Since both points have the same y-coordinate, it indicates that the quadratic inequality is horizontal and intersects the x-axis at these points.
The inequality should be in the form (x - h)^2 ≤ k, where (h, k) is the vertex.
To find the vertex, h, we can take the average of the x-coordinates from the given points: (3 + 6)/2 = 9/2 = 4.5.
To find k, we can substitute the value of h into either of the given points: (3 - 4.5)^2 = 2.25.
So, the quadratic inequality is (x - 4.5)^2 ≤ 2.25.
Now that we have the equation, we need to determine which point is a solution. We can substitute the x and y-coordinates of each point into the inequality to check.
Let's check each point:
For point (5, 9): (5 - 4.5)^2 ≤ 2.25 is false because (0.5)^2 ≤ 2.25 is false.
For point (7, 7): (7 - 4.5)^2 ≤ 2.25 is true because (2.5)^2 ≤ 2.25 is true.
Therefore, the point (7, 7) is part of the solution set.
So, the correct answer is not among the options provided.