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For the expression [tex]\frac{6x+18}{x^2-4x+4} \times \frac{x+4}{x^2-4}[/tex], which of the following are the restrictions? Select all that apply.

A. [tex]x \neq -3[/tex]
B. [tex]x \neq 2[/tex]
C. [tex]x \neq -4[/tex]
D. [tex]x \neq -2[/tex]


Sagot :

To determine the restrictions for the given expression [tex]\(\frac{6x+18}{x^2-4x+4} \times \frac{x+4}{x^2-4}\)[/tex], we need to examine the denominators and find the values of [tex]\(x\)[/tex] that would make the denominators zero, as these are the values for which the expression is undefined.

The expression has two denominators:
1. [tex]\(x^2 - 4x + 4\)[/tex]
2. [tex]\(x^2 - 4\)[/tex]

We will examine each denominator separately.

### Denominator 1: [tex]\(x^2 - 4x + 4\)[/tex]
This is a quadratic equation. We can factor it to find the roots:

[tex]\[x^2 - 4x + 4 = (x - 2)^2\][/tex]

Setting the expression equal to zero:

[tex]\[(x - 2)^2 = 0\][/tex]

Solving for [tex]\(x\)[/tex]:

[tex]\[x - 2 = 0\][/tex]

[tex]\[x = 2\][/tex]

So, [tex]\(x = 2\)[/tex] would make the first denominator zero. Thus, one restriction is [tex]\(x \neq 2\)[/tex].

### Denominator 2: [tex]\(x^2 - 4\)[/tex]
This is also a quadratic equation. We can factor it:

[tex]\[x^2 - 4 = (x - 2)(x + 2)\][/tex]

Setting the expression equal to zero:

[tex]\[(x - 2)(x + 2) = 0\][/tex]

Solving for [tex]\(x\)[/tex]:

[tex]\[x - 2 = 0 \quad \text{or} \quad x + 2 = 0\][/tex]

[tex]\[x = 2 \quad \text{or} \quad x = -2\][/tex]

So, [tex]\(x = 2\)[/tex] or [tex]\(x = -2\)[/tex] would make the second denominator zero. Thus, additional restrictions are [tex]\(x \neq 2\)[/tex] and [tex]\(x \neq -2\)[/tex].

### Summary of Restrictions
Combining the restrictions from both denominators, we have:

- [tex]\(x \neq 2\)[/tex]
- [tex]\(x \neq -2\)[/tex]

### Analysis of Given Options
The provided options are:
- [tex]\(x \neq -3\)[/tex]
- [tex]\(x \neq 2\)[/tex]
- [tex]\(x \neq -4\)[/tex]
- [tex]\(x \neq -2\)[/tex]

Based on our determined restrictions, [tex]\(x \neq 2\)[/tex] and [tex]\(x \neq -2\)[/tex] are correct.

Therefore, the restrictions for the expression are:
[tex]\[x \neq 2 \text{ and } x \neq -2\][/tex]

The correct options are:
- [tex]\(x \neq 2\)[/tex]
- [tex]\(x \neq -2\)[/tex]