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Equivalent Expressions Flashcards

6 cards from real Bluebook SAT Test practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.

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  1. Which expression is equivalent to (x² − 9)/(x² − x − 6), given that x ≠ 3 and x ≠ −2?

    Answer: (x + 3)/(x + 2)

    Factor both numerator and denominator: x² − 9 = (x − 3)(x + 3) and x² − x − 6 = (x − 3)(x + 2). Cancel the common factor (x − 3), leaving (x + 3)/(x + 2). The conditions x ≠ 3 and x ≠ −2 exclude the values that make the original expression undefined.

  2. Which expression is equivalent to (2x − 3)² − (x + 1)²?

    Answer: (3x − 2)(x − 4)

    Recognize this as a difference of squares: a² − b² = (a + b)(a − b), where a = (2x − 3) and b = (x + 1). So: [(2x − 3) + (x + 1)][(2x − 3) − (x + 1)] = (3x − 2)(x − 4). You can verify by expanding: 3x² − 12x − 2x + 8 = 3x² − 14x + 8, which matches expanding the original expression.

  3. Which of the following is equivalent to x² + 5x + 7, written in vertex form?

    Answer: (x + 5/2)² + 3/4

    Complete the square: x² + 5x + 7 = (x² + 5x + 25/4) − 25/4 + 7 = (x + 5/2)² + (−25/4 + 28/4) = (x + 5/2)² + 3/4. Choice A has the wrong sign on the constant (−3/4 instead of +3/4). Choice C uses −5/2 instead of +5/2. Choice B uses an incorrect half-coefficient.

  4. For x > 0, which expression is equivalent to x^(2/3) ÷ x^(1/6)?

    Answer: x^(1/2)

    When dividing expressions with the same base, subtract exponents: x^(2/3) ÷ x^(1/6) = x^(2/3 − 1/6). Convert to a common denominator: 2/3 = 4/6, so 4/6 − 1/6 = 3/6 = 1/2. The result is x^(1/2), or equivalently √x. A common error is adding exponents (giving 5/6) or subtracting in the wrong direction (giving 1/6).

  5. Which expression is equivalent to (1/x − 1/y) / (1/x + 1/y), for x ≠ 0, y ≠ 0, and x ≠ −y?

    Answer: (y − x)/(y + x)

    Multiply the entire complex fraction by xy/xy (multiply both numerator and denominator by xy): Numerator: (1/x − 1/y)·xy = y − x. Denominator: (1/x + 1/y)·xy = y + x. Result: (y − x)/(y + x). Choice A, (x − y)/(x + y), is the negative of the correct answer — a subtle sign flip that comes from reversing the order of terms. The difference between A and C is critical.

  6. Which expression is equivalent to (x³ + 27)/(x + 3), for x ≠ −3?

    Answer: x² − 3x + 9

    Recognize x³ + 27 as a sum of cubes: a³ + b³ = (a + b)(a² − ab + b²), where a = x and b = 3. So x³ + 27 = (x + 3)(x² − 3x + 9). Dividing by (x + 3) gives x² − 3x + 9. Choice B, x² + 3x + 9, is the most tempting wrong answer — it results from incorrectly using +ab instead of −ab in the sum of cubes formula.