To determine which values are possible for the discriminant of a quadratic function with no x-intercepts, we must understand that a quadratic function, in general form [tex]\( ax^2 + bx + c \)[/tex], has its x-intercepts determined by the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
The term under the square root, [tex]\( b^2 - 4ac \)[/tex], is known as the discriminant.
To understand the behavior of the quadratic function in relation to its x-intercepts:
1. If the discriminant [tex]\( b^2 - 4ac > 0 \)[/tex], the quadratic equation has two distinct real solutions, meaning the graph intersects the x-axis at two points.
2. If the discriminant [tex]\( b^2 - 4ac = 0 \)[/tex], the quadratic equation has exactly one real solution, which means the graph touches the x-axis at exactly one point (vertex of the parabola on the x-axis).
3. If the discriminant [tex]\( b^2 - 4ac < 0 \)[/tex], the quadratic equation has no real solutions, meaning the graph does not intersect the x-axis at any point.
Given the problem, we need to identify the values of the discriminant such that the quadratic function has no x-intercepts. Hence, we are interested in the values where the discriminant [tex]\( b^2 - 4ac < 0 \)[/tex].
Examining the given options:
- A. -18 (discriminant is less than 0)
- B. 3 (discriminant is greater than 0)
- C. -1 (discriminant is less than 0)
- D. 0 (discriminant equals 0)
From the above criteria, we select the values where the discriminant is less than 0. Therefore, the possible values for the discriminant are:
A. -18
C. -1
Thus, the discriminant values that ensure the quadratic function has no x-intercepts are [tex]\(-18\)[/tex] and [tex]\(-1\)[/tex].
