Question
on the set of aces below solve the following system of equations graphically and steate the coordinates of all points in the soloution set
y=x^2+10x+24
6x+3y=-24
you can move the parabola by moving the dots state the x intercepts
y=x^2+10x+24
6x+3y=-24
you can move the parabola by moving the dots state the x intercepts
Answers
GPT-4o mini
To solve the system of equations graphically and find the coordinates of all points in the solution set, let's first analyze both equations:
1. **Quadratic Equation:**
\[
y = x^2 + 10x + 24
\]
This is a parabola that opens upwards.
2. **Linear Equation:**
\[
6x + 3y = -24
\]
To express this in slope-intercept form, we can rearrange it:
\[
3y = -6x - 24 \quad \Rightarrow \quad y = -2x - 8
\]
This is a straight line with a slope of -2 and a y-intercept of -8.
### Finding the x-intercepts of the Parabola
To find the x-intercepts of the quadratic equation, we set \(y = 0\):
\[
0 = x^2 + 10x + 24
\]
Now we can factor the quadratic:
\[
0 = (x + 6)(x + 4)
\]
Setting each factor to zero gives:
\[
x + 6 = 0 \quad \Rightarrow \quad x = -6
\]
\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\]
Thus, the x-intercepts are at:
\[
(-6, 0) \quad \text{and} \quad (-4, 0)
\]
### Graphing the Equations
To graph these equations, you would:
1. **Graph the quadratic \(y = x^2 + 10x + 24\)**:
- It opens upwards, and its vertex can be found using \(x = -\frac{b}{2a} = -\frac{10}{2} = -5\). Substituting \(x = -5\) into the equation gives \(y = 1\), which is the vertex.
2. **Graph the linear equation \(y = -2x - 8\)**:
- This is a straight line that crosses the y-axis at (0, -8) and has a slope of -2, indicating it goes downwards.
### Finding Intersection Points
To find the intersection points, we set the two equations equal to each other:
\[
x^2 + 10x + 24 = -2x - 8
\]
Rearranging gives:
\[
x^2 + 12x + 32 = 0
\]
Now we factor or use the quadratic formula:
\[
x^2 + 12x + 32 = (x + 4)(x + 8) = 0
\]
Setting each factor to zero gives:
\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\]
\[
x + 8 = 0 \quad \Rightarrow \quad x = -8
\]
### Finding Corresponding y-values
1. For \(x = -4\):
\[
y = -2(-4) - 8 = 8 - 8 = 0
\]
The point is \((-4, 0)\).
2. For \(x = -8\):
\[
y = -2(-8) - 8 = 16 - 8 = 8
\]
The point is \((-8, 8)\).
### Conclusion
**The coordinates of the intersection points are:**
1. \((-4, 0)\)
2. \((-8, 8)\)
### X-intercepts of the Parabola
The x-intercepts of the parabola \(y = x^2 + 10x + 24\) are:
- \((-6, 0)\)
- \((-4, 0)\)
Thus, we have found all necessary points and the x-intercepts. The solution set consists of the intersection points: \((-4, 0)\) and \((-8, 8)\).
1. **Quadratic Equation:**
\[
y = x^2 + 10x + 24
\]
This is a parabola that opens upwards.
2. **Linear Equation:**
\[
6x + 3y = -24
\]
To express this in slope-intercept form, we can rearrange it:
\[
3y = -6x - 24 \quad \Rightarrow \quad y = -2x - 8
\]
This is a straight line with a slope of -2 and a y-intercept of -8.
### Finding the x-intercepts of the Parabola
To find the x-intercepts of the quadratic equation, we set \(y = 0\):
\[
0 = x^2 + 10x + 24
\]
Now we can factor the quadratic:
\[
0 = (x + 6)(x + 4)
\]
Setting each factor to zero gives:
\[
x + 6 = 0 \quad \Rightarrow \quad x = -6
\]
\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\]
Thus, the x-intercepts are at:
\[
(-6, 0) \quad \text{and} \quad (-4, 0)
\]
### Graphing the Equations
To graph these equations, you would:
1. **Graph the quadratic \(y = x^2 + 10x + 24\)**:
- It opens upwards, and its vertex can be found using \(x = -\frac{b}{2a} = -\frac{10}{2} = -5\). Substituting \(x = -5\) into the equation gives \(y = 1\), which is the vertex.
2. **Graph the linear equation \(y = -2x - 8\)**:
- This is a straight line that crosses the y-axis at (0, -8) and has a slope of -2, indicating it goes downwards.
### Finding Intersection Points
To find the intersection points, we set the two equations equal to each other:
\[
x^2 + 10x + 24 = -2x - 8
\]
Rearranging gives:
\[
x^2 + 12x + 32 = 0
\]
Now we factor or use the quadratic formula:
\[
x^2 + 12x + 32 = (x + 4)(x + 8) = 0
\]
Setting each factor to zero gives:
\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\]
\[
x + 8 = 0 \quad \Rightarrow \quad x = -8
\]
### Finding Corresponding y-values
1. For \(x = -4\):
\[
y = -2(-4) - 8 = 8 - 8 = 0
\]
The point is \((-4, 0)\).
2. For \(x = -8\):
\[
y = -2(-8) - 8 = 16 - 8 = 8
\]
The point is \((-8, 8)\).
### Conclusion
**The coordinates of the intersection points are:**
1. \((-4, 0)\)
2. \((-8, 8)\)
### X-intercepts of the Parabola
The x-intercepts of the parabola \(y = x^2 + 10x + 24\) are:
- \((-6, 0)\)
- \((-4, 0)\)
Thus, we have found all necessary points and the x-intercepts. The solution set consists of the intersection points: \((-4, 0)\) and \((-8, 8)\).