Question

Which of the following expressions simplifies to an equivalent simplified form of
x
2
−4
x
2
−x−2
​
?
A.
x
2
+x−2
x
2
−1
​

B.
x+2
x
2
+3x+2
​

C.
x
2
−x−2
x
2
−4
​

D.
x
2
−4
x
2
−2x
​

Answer 1

To simplify the expression \( \frac{x^2 - 4}{x^2 - x - 2} \), we will first factor both the numerator and the denominator.

1. **Factor the numerator:**
\[
x^2 - 4 = (x - 2)(x + 2)
\]

2. **Factor the denominator:**
\[
x^2 - x - 2 = (x - 2)(x + 1)
\]

Now we can rewrite the expression as:

\[
\frac{(x - 2)(x + 2)}{(x - 2)(x + 1)}
\]

We can cancel the common factor \( (x - 2) \) (assuming \( x \neq 2 \)):

\[
\frac{x + 2}{x + 1}
\]

Now we will examine each option to see which one simplifies to this result.

**A. \( \frac{x^2 + x - 2}{x^2 - 1} \)**

Factoring the numerator \( x^2 + x - 2 = (x - 1)(x + 2) \) and the denominator \( x^2 - 1 = (x - 1)(x + 1) \):

\[
\frac{(x - 1)(x + 2)}{(x - 1)(x + 1)} \quad \text{(assuming \( x \neq 1 \))}
\]

Cancelling \( (x - 1) \):

\[
\frac{x + 2}{x + 1}
\]

So A simplifies to \( \frac{x + 2}{x + 1} \).

**B. \( \frac{x + 2}{x^2 + 3x + 2} \)**

Factoring the denominator \( x^2 + 3x + 2 = (x + 1)(x + 2) \):

\[
\frac{x + 2}{(x + 1)(x + 2)} \quad \text{(assuming \( x \neq -2 \))}
\]

Cancelling \( (x + 2) \):

\[
\frac{1}{x + 1}
\]

B does not simplify to \( \frac{x + 2}{x + 1} \).

**C. \( \frac{x^2 - x - 2}{x^2 - 4} \)**

We previously factored both:

Numerator: \( x^2 - x - 2 = (x - 2)(x + 1) \)
Denominator: \( x^2 - 4 = (x - 2)(x + 2) \)

The expression becomes:

\[
\frac{(x - 2)(x + 1)}{(x - 2)(x + 2)} \quad \text{(assuming \( x \neq 2 \))}
\]

Cancelling \( (x - 2) \):

\[
\frac{x + 1}{x + 2}
\]

C does not simplify to \( \frac{x + 2}{x + 1} \).

**D. \( \frac{x^2 - 4}{x^2 - 2x} \)**

Factoring numerator: \( x^2 - 4 = (x - 2)(x + 2) \)
Factoring denominator: \( x^2 - 2x = x(x - 2) \)

The expression becomes:

\[
\frac{(x - 2)(x + 2)}{x(x - 2)} \quad \text{(assuming \( x \neq 2 \))}
\]

Cancelling \( (x - 2) \):

\[
\frac{x + 2}{x}
\]

D does not simplify to \( \frac{x + 2}{x + 1} \).

Thus, the only expression that simplifies to an equivalent form of \(\frac{x^2 - 4}{x^2 - x - 2}\) is **A.**