For x 2 and x -3, which expression is equivalent to x^2 - 4x^2 + x - 6?

For x 2 and x -3, which expression is equivalent to x^2 -... | SAT Equivalent expressions Question, Answer Explained

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Math · Advanced MathHard

For $x \ne 2$ and $x \ne -3$ , which expression is equivalent to $\frac{x^2 - 4}{x^2 + x - 6}$ ?

$\frac{-4}{x - 6}$

$\frac{x - 2}{x - 3}$

$\frac{x + 2}{x - 3}$

$\frac{x + 2}{x + 3}$

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Answer: D

Factor the numerator and the denominator: $x^2 - 4 = (x + 2)(x - 2)$ and $x^2 + x - 6 = (x + 3)(x - 2)$ . Cancel the common factor $x - 2$ to get $\frac{x + 2}{x + 3}$ .

Choice A cancels the $x^2$ terms, but only common factors can be canceled, not terms that are added or subtracted.

Why the other choices are wrong

A. This cancels the x² terms. Only common factors can be canceled, not terms in a sum.
B. This factors x² + x - 6 as (x + 2)(x - 3), which equals x² - x - 6. The correct factors are (x + 3)(x - 2).
C. The denominator factors as (x + 3)(x - 2), not (x - 3)(x - 2). After canceling x - 2, the denominator is x + 3.

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