To understand what the zeros of the quadratic equation [tex]\(x^2 + 4x = -2x + 16\)[/tex] tell us about its graph, let's start by rewriting the equation and analyzing its solution.
First, simplify and standardize the quadratic equation:
[tex]\[
x^2 + 4x + 2x - 16 = 0
\][/tex]
Combine like terms:
[tex]\[
x^2 + 6x - 16 = 0
\][/tex]
To find the zeros of the quadratic equation, we solve [tex]\(x^2 + 6x - 16 = 0\)[/tex]. The solutions to this equation are the values of [tex]\(x\)[/tex] where the quadratic function intersects the x-axis.
The zeros (roots) of this quadratic function are [tex]\(x = -8\)[/tex] and [tex]\(x = 2\)[/tex].
1. Zeros or Roots: The solutions, [tex]\(x = -8\)[/tex] and [tex]\(x = 2\)[/tex], are the x-intercepts of the parabola. This means the parabola crosses the x-axis at these points.
2. Graph Behavior: The x-intercepts tell us critical points where the graph touches or crosses the x-axis.
Based on this information, let's discuss each of the provided statements:
- The parabola crosses the x-axis at [tex]\(x = -8\)[/tex] and [tex]\(x = 2\)[/tex]:
This statement is correct. Since both [tex]\(x = -8\)[/tex] and [tex]\(x = 2\)[/tex] are zeros of the quadratic equation, the parabola crosses the x-axis at these two points.
- The parabola crosses the x-axis at [tex]\(x = -8\)[/tex] and [tex]\(x = 2\)[/tex], but you can ignore the zero [tex]\(x = -8\)[/tex] because it is negative:
This statement is incorrect. Both zeros are important in understanding the behavior of the parabola. Ignoring one of them because it's negative is not a valid mathematical practice.
- The parabola does not touch or cross the x-axis:
This statement is incorrect. The zeros [tex]\(x = -8\)[/tex] and [tex]\(x = 2\)[/tex] specifically indicate the points where the parabola touches or crosses the x-axis.
- The parabola touches the x-axis only at [tex]\(x = 2\)[/tex]:
This statement is incorrect. The parabola touches or crosses the x-axis at both [tex]\(x = -8\)[/tex] and [tex]\(x = 2\)[/tex].
Therefore, the correct conclusion is:
The parabola crosses the x-axis at [tex]\(x = -8\)[/tex] and [tex]\(x = 2\)[/tex].
