To determine the locations where the bumper cars will hit one another, we need to find the points of intersection between the two curves given by the equations [tex]\(2x + y = -4\)[/tex] and [tex]\(y = x^2 - x - 6\)[/tex].
### Step-by-Step Solution:
1. Rewrite the equations if necessary:
- First equation: [tex]\(2x + y = -4\)[/tex]
- Second equation: [tex]\(y = x^2 - x - 6\)[/tex]
2. Substitute the expression for [tex]\(y\)[/tex] from the second equation into the first equation:
- From the second equation: [tex]\(y = x^2 - x - 6\)[/tex]
- Substitute [tex]\(y\)[/tex] in the first equation:
[tex]\[
2x + (x^2 - x - 6) = -4
\][/tex]
- Simplify the equation:
[tex]\[
2x + x^2 - x - 6 = -4
\][/tex]
[tex]\[
x^2 + x - 6 = -4
\][/tex]
3. Solve the quadratic equation:
- Move all terms to one side to set the equation to zero:
[tex]\[
x^2 + x - 6 + 4 = 0
\][/tex]
[tex]\[
x^2 + x - 2 = 0
\][/tex]
- Factor the quadratic equation:
[tex]\[
(x + 2)(x - 1) = 0
\][/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\[
x + 2 = 0 \quad \text{or} \quad x - 1 = 0
\][/tex]
[tex]\[
x = -2 \quad \text{or} \quad x = 1
\][/tex]
4. Find the corresponding [tex]\(y\)[/tex] values:
- For [tex]\(x = -2\)[/tex]:
[tex]\[
y = (-2)^2 - (-2) - 6
\][/tex]
[tex]\[
y = 4 + 2 - 6 = 0
\][/tex]
So, one intersection point is [tex]\((-2, 0)\)[/tex].
- For [tex]\(x = 1\)[/tex]:
[tex]\[
y = 1^2 - 1 - 6
\][/tex]
[tex]\[
y = 1 - 1 - 6 = -6
\][/tex]
So, the other intersection point is [tex]\((1, -6)\)[/tex].
### Conclusion
The bumper cars will hit each other at the points [tex]\((-2, 0)\)[/tex] and [tex]\((1, -6)\)[/tex].
