To find the [tex]$x$[/tex]-intercepts of the function [tex]\( n(x) = x(4x - 12) \)[/tex], follow these steps:
1. Understand what an [tex]\( x \)[/tex]-intercept is:
An [tex]\( x \)[/tex]-intercept is a point where the graph of the function crosses the [tex]\( x \)[/tex]-axis. At these points, the [tex]\( y \)[/tex]-coordinate (or [tex]\( n(x) \)[/tex]) is equal to 0.
2. Set the function equal to 0:
To find the [tex]\( x \)[/tex]-intercepts, we need to solve the equation [tex]\( n(x) = 0 \)[/tex].
Given the function [tex]\( n(x) = x(4x - 12) \)[/tex], set [tex]\( n(x) \)[/tex] to 0:
[tex]\[
x(4x - 12) = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
We can solve the equation by using the zero product property, which states that if [tex]\( ab = 0 \)[/tex], then either [tex]\( a = 0 \)[/tex] or [tex]\( b = 0 \)[/tex].
Here, we have [tex]\( x(4x - 12) = 0 \)[/tex], so:
[tex]\[
x = 0 \quad \text{or} \quad 4x - 12 = 0
\][/tex]
- For [tex]\( x = 0 \)[/tex]
- For [tex]\( 4x - 12 = 0 \)[/tex]:
[tex]\[
4x - 12 = 0
\][/tex]
Add 12 to both sides:
[tex]\[
4x = 12
\][/tex]
Divide both sides by 4:
[tex]\[
x = 3
\][/tex]
4. Identify the [tex]\( x \)[/tex]-intercepts:
The solutions to the equation tell us the [tex]\( x \)[/tex]-coordinates of the [tex]\( x \)[/tex]-intercepts:
- When [tex]\( x = 0 \)[/tex], [tex]\( n(x) = 0 \)[/tex], giving the point [tex]\( (0, 0) \)[/tex].
- When [tex]\( x = 3 \)[/tex], [tex]\( n(x) = 0 \)[/tex], giving the point [tex]\( (3, 0) \)[/tex].
5. Write the [tex]\( x \)[/tex]-intercepts in coordinate form:
The [tex]\( x \)[/tex]-intercepts of the function [tex]\( n(x) \)[/tex] are:
[tex]\[
(0, 0) \quad \text{and} \quad (3, 0)
\][/tex]
So, the [tex]\( x \)[/tex]-intercepts of the graph of the function [tex]\( n(x) = x(4x - 12) \)[/tex] are [tex]\((0, 0)\)[/tex] and [tex]\((3, 0)\)[/tex].
