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What is the quotient?

[tex]\[ \frac{x^2-4}{(x-2)(x-4)} \][/tex]

A. [tex]\(\frac{x+2}{x-4}\)[/tex]

B. [tex]\(\frac{x-2}{x-4}\)[/tex]

C. [tex]\(\frac{x}{x-2}\)[/tex]

D. [tex]\(\frac{x+2}{x-2}\)[/tex]

Sagot :

GinnyAnsweridn
To determine the quotient of the given expression:

[tex]\[ \frac{x^2-4}{(x-2)(x-4)} \][/tex]

we need to simplify it step-by-step.

1. Factor the numerator: The numerator [tex]\( x^2 - 4 \)[/tex] is a difference of squares, which we can factor as follows:

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

2. Rewrite the expression with the factored numerator:

[tex]\[ \frac{(x + 2)(x - 2)}{(x - 2)(x - 4)} \][/tex]

3. Cancel common factors: We see that [tex]\( (x - 2) \)[/tex] is present in both the numerator and the denominator. So, we can cancel out [tex]\( (x - 2) \)[/tex]:

[tex]\[ \frac{(x + 2) \cancel{(x - 2)}}{\cancel{(x - 2)}(x - 4)} = \frac{x + 2}{x - 4} \][/tex]

Therefore, the simplified form of the given expression is:

[tex]\[ \frac{x + 2}{x - 4} \][/tex]

Among the provided options, this matches option A:

[tex]\[ \boxed{\frac{x+2}{x-4}} \][/tex]

Thus, the correct answer is:

A. [tex]\(\frac{x+2}{x-4}\)[/tex]
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