Question

on the set of axes below, solve the following system of equations graphically and state the coordinates of all points in the solution set.
$y = x^2 + 10x + 24$
$6x + 3y = -24$
you can move the parabola by dragging the dots. graph the line by clicking twice.

Explanation:

Step1: Analyze the parabola \( y = x^2 + 10x + 24 \)

First, we can factor the quadratic function: \( y=(x + 4)(x+6) \). So the roots of the parabola (where \( y = 0 \)) are \( x=-4 \) and \( x=-6 \). The vertex of the parabola \( y = ax^2+bx + c \) is at \( x=-\frac{b}{2a} \). For \( y=x^2 + 10x + 24 \), \( a = 1 \), \( b = 10 \), so the x - coordinate of the vertex is \( x=-\frac{10}{2\times1}=-5 \). Substituting \( x = - 5 \) into the equation, we get \( y=(-5)^2+10\times(-5)+24=25 - 50 + 24=-1 \). So the vertex of the parabola is at \( (-5,-1) \).

Step2: Analyze the line \( 6x + 3y=-24 \)

We can rewrite this equation in slope - intercept form (\( y=mx + b \)) by solving for \( y \).
Divide the entire equation by 3: \( 2x + y=-8 \), then \( y=-2x - 8 \).
To graph the line, we can find two points on the line.
When \( x = 0 \), \( y=-8 \). So one point is \( (0,-8) \).
When \( y = 0 \), \( 0=-2x - 8\), then \( 2x=-8 \), \( x=-4 \). So another point is \( (-4,0) \).

Step3: Find the intersection points (graphically)

The parabola \( y=x^2 + 10x + 24 \) and the line \( y=-2x - 8 \) intersect where \( x^2+10x + 24=-2x - 8 \).
Rearranging this equation gives \( x^2+12x + 32 = 0 \).
Factor the quadratic equation: \( x^2+12x + 32=(x + 4)(x + 8)=0 \).
So \( x=-4 \) or \( x=-8 \).

• When \( x=-4 \), substitute into the line equation \( y=-2\times(-4)-8 = 8 - 8=0 \).
• When \( x=-8 \), substitute into the line equation \( y=-2\times(-8)-8=16 - 8 = 8 \).

We can also verify by looking at the graphs. The parabola has roots at \( x=-4 \) and \( x=-6 \), vertex at \( (-5,-1) \). The line passes through \( (0,-8) \) and \( (-4,0) \). By graphing both, we can see the intersection points.

Answer:

The solution set of the system of equations (the coordinates of the intersection points) is \( (-4,0) \) and \( (-8,8) \)