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What is the midpoint of the [tex]\(x\)[/tex]-intercepts of [tex]\(f(x) = (x-2)(x-4)\)[/tex]?

A. [tex]\((-3, 0)\)[/tex]

B. [tex]\((-1, 0)\)[/tex]

C. [tex]\((1, 0)\)[/tex]

D. [tex]\((3, 0)\)[/tex]


Sagot :

To find the midpoint of the [tex]\( x \)[/tex]-intercepts of the function [tex]\( f(x) = (x-2)(x-4) \)[/tex], follow these steps:

1. Identify the [tex]\( x \)[/tex]-intercepts:
The [tex]\( x \)[/tex]-intercepts of a function occur where the function equals zero. To find them, we set [tex]\( f(x) = 0 \)[/tex] and solve the equation:
[tex]\[ (x - 2)(x - 4) = 0 \][/tex]

2. Solve for [tex]\( x \)[/tex]:
This equation is satisfied when either [tex]\( x - 2 = 0 \)[/tex] or [tex]\( x - 4 = 0 \)[/tex]. Solving these:
[tex]\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \][/tex]
[tex]\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are at [tex]\( x = 2 \)[/tex] and [tex]\( x = 4 \)[/tex].

3. Find the midpoint of the [tex]\( x \)[/tex]-intercepts:
To find the midpoint of these points, we use the midpoint formula for two points, [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex], which is given by:
[tex]\[ \text{Midpoint} = \frac{x_1 + x_2}{2} \][/tex]
Here, [tex]\( x_1 = 2 \)[/tex] and [tex]\( x_2 = 4 \)[/tex]. Substituting these values into the formula:
[tex]\[ \text{Midpoint} = \frac{2 + 4}{2} = \frac{6}{2} = 3 \][/tex]

4. Result:
Therefore, the midpoint of the [tex]\( x \)[/tex]-intercepts is [tex]\( 3 \)[/tex].

Given the options provided, the correct answer is:
[tex]\[ (3,0) \][/tex]
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