Final answer:
The given quadratic equation X² - 7x + 21 = 0 has a vertex at x = 3.5, which represents the axis of symmetry. The vertex corresponds to a minimum value because the parabola opens upward due to a positive coefficient of the X² term.
Step-by-step explanation:
The question is about solving a quadratic equation and finding its vertex, which in turn helps in determining the axis of symmetry (AOS) and whether the vertex represents a maximum or a minimum value on the graph of the quadratic function.
To begin with, we should first rewrite the given equation X²− 10x + 21 = −3x into standard form ax² + bx + c = 0 by adding 3x to both sides, resulting in X² - 7x + 21 = 0. The next step is to calculate the AOS, which is given by -b/(2a). In the case of our equation, 'a' is 1, and 'b' is -7. Thus AOS is 7/2 or 3.5. This x-value also corresponds to the vertex of the parabola.
To determine whether the parabola opens up or down, we look at the coefficient of the X² term, which is positive, indicating that the parabola opens up and thus has a minimum value at its vertex. Therefore, the vertex represents a minimum point on the graph.