To find the solutions, we need to determine both the [tex]$x$[/tex]-intercepts and the vertex of the quadratic function [tex]\( y = (x + 4)(x - 2) \)[/tex].
### Finding the [tex]$x$[/tex]-intercepts:
To find the [tex]$x$[/tex]-intercepts, we set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex] in the equation [tex]\( (x + 4)(x - 2) = 0 \)[/tex].
Set each factor equal to zero:
[tex]\[ x + 4 = 0 \implies x = -4 \][/tex]
[tex]\[ x - 2 = 0 \implies x = 2 \][/tex]
So, the [tex]$x$[/tex]-intercepts are [tex]\((-4, 0)\)[/tex] and [tex]\((2, 0)\)[/tex].
### Finding the vertex:
The vertex can be found using the standard form of a quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex]. We expand the given equation [tex]\( y = (x + 4)(x - 2) \)[/tex] first:
[tex]\[ y = x^2 - 2x + 4x - 8 \][/tex]
[tex]\[ y = x^2 + 2x - 8 \][/tex]
From this expanded form, we identify [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -8 \)[/tex].
The vertex of a parabola in the form [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x = -\frac{2}{2 \cdot 1} = -1 \][/tex]
To find the y-coordinate of the vertex, substitute [tex]\( x = -1 \)[/tex] back into the equation [tex]\( y = x^2 + 2x - 8 \)[/tex]:
[tex]\[ y = (-1)^2 + 2(-1) - 8 \][/tex]
[tex]\[ y = 1 - 2 - 8 \][/tex]
[tex]\[ y = -9 \][/tex]
So, the vertex is [tex]\((-1, -9)\)[/tex].
### Conclusion:
- The solution for the x-intercepts is [tex]\((x = -4, 0)\)[/tex] and [tex]\((x = 2, 0)\)[/tex]. Thus, the correct answer for the [tex]$x$[/tex]-intercepts is:
C. [tex]$x$[/tex]-intercepts: [tex]\((-4,0),(2,0)\)[/tex]
- The solution for the vertex is [tex]\((-1, -9)\)[/tex]. Thus, the correct answer for the vertex is:
D. Vertex: [tex]\((-1,-9)\)[/tex]
