To solve the quadratic equation [tex]\( x^2 - 4x - 32 = 0 \)[/tex] by graphing, follow these steps:
1. Identify the Parabola: The quadratic equation [tex]\( x^2 - 4x - 32 = 0 \)[/tex] represents a parabola. The general form of the quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex] where [tex]\( a = 1 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = -32 \)[/tex].
2. Find the Vertex: The vertex of a parabola defined by [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
[tex]\[
x = -\frac{-4}{2 \cdot 1} = \frac{4}{2} = 2
\][/tex]
Substitute [tex]\( x = 2 \)[/tex] back into the equation to find the corresponding [tex]\( y \)[/tex]-coordinate.
[tex]\[
y = (2)^2 - 4(2) - 32 = 4 - 8 - 32 = -36
\][/tex]
So, the vertex of the parabola is at [tex]\( (2, -36) \)[/tex].
3. Determine the Roots: The roots of the quadratic function are the values of [tex]\( x \)[/tex] where the function equals zero. These are commonly referred to as the x-intercepts of the parabola. Given the quadratic equation [tex]\( x^2 - 4x - 32 = 0 \)[/tex], we need to find the points where the graph of the equation intersects the x-axis ([tex]\( y = 0 \)[/tex]).
4. Solve the Quadratic Equation: The x-intercepts or roots can be found using factoring, the quadratic formula, or graphing. For our purposes:
[tex]\[
x^2 - 4x - 32 = 0
\][/tex]
Solving for [tex]\( x \)[/tex] yields the roots:
[tex]\[
x = -4 \quad \text{and} \quad x = 8
\][/tex]
5. Graph the Parabola: By plotting the vertex [tex]\( (2, -36) \)[/tex] and the roots [tex]\( (-4, 0) \)[/tex] and [tex]\( (8, 0) \)[/tex], you can visualize the parabola opening upwards, intersecting the x-axis at these points.
6. Identify the Correct Answer: By observing where the parabola intersects the x-axis, we can determine the correct pair of x-intercepts.
From the given options:
- a. [tex]\((-4.0)\)[/tex] and [tex]\((-8,0)\)[/tex]
- b. [tex]\((0,8)\)[/tex] and [tex]\((0,4)\)[/tex]
- c. [tex]\((8,0)\)[/tex] and [tex]\((-4,0)\)[/tex]
- d. [tex]\((0,-4)\)[/tex] and [tex]\((0,-8)\)[/tex]
The correct choice that lists the x-intercepts is:
c. [tex]\((8,0)\)[/tex] and [tex]\((-4,0)\)[/tex].
