ACT Math Quiz: Equivalent Expressions
20 questions · exam conditions
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Equivalent ExpressionsQuestion 1 of 20

Which of the following expressions is equivalent to 3(x−4)+2x3(x - 4) + 2x?

5x−45x - 4
5x−125x - 12
6x−126x - 12
x−12x - 12
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ACT Math Quiz

ACT Math Quiz: Equivalent Expressions

Practice Equivalent Expressions in ACT Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Equivalent Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for ACT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Which of the following expressions is equivalent to 3(x−4)+2x3(x - 4) + 2x?

  1. 5x−45x - 4
  2. 5x−125x - 12 (correct answer)
  3. 6x−126x - 12
  4. x−12x - 12
Explanation: The correct answer is B (5x − 12). Distribute the 3 across the parentheses: 3(x − 4) = 3x − 12. Then combine like terms with 2x: 3x − 12 + 2x = 5x − 12. A (5x − 4) results from distributing 3 to x but not to −4, keeping −4 instead of computing 3 × (−4) = −12. C (6x − 12) comes from incorrectly treating the 2x as adding to the coefficient 3 rather than to 3x. D (x − 12) results from subtracting 2x rather than adding it. Distribution errors are extremely common — always multiply the outside factor by every term inside the parentheses.

Question 2

For all xx and yy, which of the following expressions is equivalent to (3x3y2)(4xy4)(3x^3y^2)(4xy^4)?

  1. 7x3y67x^3y^6
  2. 12x3y812x^3y^8
  3. 12x4y612x^4y^6 (correct answer)
  4. 7x4y67x^4y^6
Explanation: The correct answer is C (12x⁴y⁶). Multiply the coefficients: 3 × 4 = 12. Apply the product rule for exponents (add exponents of like bases): x³ × x¹ = x^(3+1) = x⁴; y² × y⁴ = y^(2+4) = y⁶. Result: 12x⁴y⁶. A (7x³y⁶) adds coefficients (3+4=7) instead of multiplying and doesn't add the x exponent. B (12x³y⁸) correctly multiplies coefficients but treats x³ as fixed and incorrectly adds the y exponents (possibly counting y⁴ twice). D (7x⁴y⁶) adds coefficients but correctly adds the exponents. Key rule: multiply coefficients, add exponents of like bases.

Question 3

A student rewrites an expression while simplifying. What is the simplified form of 7x−3+2x+57x - 3 + 2x + 5?

  1. 9x−29x - 2
  2. 9x−89x - 8
  3. 9x+29x + 2 (correct answer)
  4. 5x+25x + 2
Explanation: Simplifying means combining like terms, and each term travels with the sign in front of it. The xx terms give 7x+2x=9x7x + 2x = 9x, and the constants give −3+5=2-3 + 5 = 2, so the expression simplifies to 9x+29x + 2. The answer 9x−29x - 2 subtracts the constants as 3−53 - 5 instead of adding −3-3 and 55; 9x−89x - 8 treats the +5+5 as −5-5 and computes −3−5-3 - 5; and 5x+25x + 2 subtracts the variable terms as 7x−2x7x - 2x even though the 2x2x is being added. Before combining, rewrite the expression as a sum such as 7x+(−3)+2x+57x + (-3) + 2x + 5 so every sign is attached to its own term and cannot drift.

Question 4

Which expression is equivalent to 4x(2+3)4x(2 + 3)?

  1. 1010
  2. 20x20x (correct answer)
  3. 10x10x
  4. 2020
Explanation: Order of operations says to simplify inside the parentheses before multiplying, and the variable outside stays attached through the multiplication. Inside, 2+3=52 + 3 = 5, so the expression becomes 4x⋅5=20x4x \cdot 5 = 20x; distributing gives the same result, since 4x(2)+4x(3)=8x+12x=20x4x(2) + 4x(3) = 8x + 12x = 20x. The value 2020 carries out the multiplication correctly but drops the variable, which cannot disappear because nothing cancels it. Both 10x10x and 1010 come from multiplying the 5 by the 2 inside the parentheses instead of by the 4 outside, and 1010 additionally loses the xx. Whenever a coefficient with a variable multiplies a numerical sum, collapse the sum first and keep the variable on the product.

Question 5

Which expression is equivalent to 3x(2+5)3x(2 + 5)?

  1. 6x+156x+15
  2. 21x21x (correct answer)
  3. 7x7x
  4. 2121
Explanation: Simplify inside the parentheses first, then multiply, keeping the variable attached to the product. Since 2+5=72 + 5 = 7, the expression becomes 3x⋅7=21x3x \cdot 7 = 21x, and distributing agrees, because 3x(2)+3x(5)=6x+15x=21x3x(2) + 3x(5) = 6x + 15x = 21x. The value 2121 does the multiplication correctly but loses the xx, which nothing in the problem cancels. The expression 6x+156x+15 distributes but drops the variable from the second product, since 3x(5)3x(5) is 15x15x, not 1515, and 7x7x comes from adding inside the parentheses and then forgetting to multiply by the 3. When a term like 3x3x multiplies a numerical sum, every piece of the product keeps the variable, so a variable-free term signals a lost factor.

Question 6

Simplify: 6(m−2)+4(m+1)6(m - 2) + 4(m + 1)

  1. $10m - 2$
  2. 10m+210m + 2
  3. $10m - 8$ (correct answer)
  4. 10m+1010m + 10
Explanation: To simplify this expression, we need to apply the distributive property to both terms. First, distribute: 6(m - 2) = 6m - 12 and 4(m + 1) = 4m + 4. The expression becomes 6m - 12 + 4m + 4. Combining like terms: (6m + 4m) + (-12 + 4) = 10m - 8. Choice B incorrectly has +2 instead of -8 as the constant term.

Question 7

What is the simplified form of 7x−3(2x−4)7x - 3(2x - 4)?

  1. 7x−6x+127x - 6x + 12
  2. x+12x + 12 (correct answer)
  3. 13x−1213x - 12
  4. x−12x - 12
Explanation: To simplify this expression, we need to apply the distributive property. First, distribute the -3 to both terms in the parentheses: -3(2x - 4) = -6x + 12. The expression becomes 7x - 6x + 12. Combining like terms: (7x - 6x) + 12 = x + 12. Choice C incorrectly adds the coefficients of x terms instead of subtracting.

Question 8

Which expression is equivalent to 2x2(3x)2x^2(3x)?

  1. 6x66x^6
  2. 6x56x^5
  3. 5x25x^2
  4. 6x36x^3 (correct answer)
Explanation: Multiply the coefficients and add the exponents when multiplying powers with the same base. 2x² × 3x = (2 × 3)(x² × x) = 6x³. The exponents add: x² × x¹ = x²⁺¹ = x³. Choice A incorrectly multiplies the exponents instead of adding them.

Question 9

Which of the following expressions is equivalent to 4(a+2b)−2(a−3b)4(a + 2b) - 2(a - 3b)?

  1. 2a+2b2a + 2b
  2. 2a+5b2a + 5b
  3. 6a+14b6a + 14b
  4. 2a+14b2a + 14b (correct answer)
Explanation: This is a distribution and simplification question testing the distributive property with negatives. Choice D (2a + 14b) is correct — distribute the 4: 4a + 8b. Distribute the −2 across (a − 3b): −2a + 6b. Note: −2 × (−3b) = +6b, not −6b. Combine like terms: (4a − 2a) + (8b + 6b) = 2a + 14b. Choice A (2a + 2b) results from treating the second distribution as −2(a − 3b) = −2a − 6b (wrong sign on 3b), giving 8b − 6b = 2b. Choice B (2a + 5b) is an arithmetic error in combining the b terms, possibly computing 8b − 3b. Choice C (6a + 14b) adds 4a + 2a = 6a instead of subtracting, getting the sign wrong on the a-coefficient of the second term. Pro tip: When distributing a negative number, BOTH terms inside the parentheses change sign. Write out −2(a − 3b) = −2a + 6b before combining anything.

Question 10

Factor: x2−4x−5x^2 - 4x - 5

  1. (x+5)(x+1)(x + 5)(x + 1)
  2. (x−1)(x+5)(x - 1)(x + 5)
  3. (x−5)(x−1)(x - 5)(x - 1)
  4. (x−5)(x+1)(x - 5)(x + 1) (correct answer)
Explanation: To factor this quadratic expression, find two numbers that multiply to -5 (the constant term) and add to -4 (the coefficient of x). The numbers -5 and 1 satisfy both conditions: (-5) × 1 = -5 and (-5) + 1 = -4. Therefore, x² - 4x - 5 = (x - 5)(x + 1). You can verify by expanding: (x - 5)(x + 1) = x² + x - 5x - 5 = x² - 4x - 5.

Question 11

Which expression is equivalent to x(x−7)+3xx(x - 7) + 3x?

  1. x2−4xx^2 - 4x (correct answer)
  2. x2−10xx^2 - 10x
  3. x2+4xx^2 + 4x
  4. x2−7x^2 - 7
Explanation: First distribute x through the parentheses: x(x - 7) = x² - 7x. Then add 3x to get: x² - 7x + 3x = x² - 4x. Combine like terms: -7x + 3x = -4x, while the x² term remains unchanged. Choice B incorrectly shows -10x instead of -4x.

Question 12

Which of the following matches the expansion of (x−3)(x+2)(x - 3)(x + 2)?

  1. x2−x−6x^2 - x - 6 (correct answer)
  2. x2+5x−6x^2 + 5x - 6
  3. x2−6x^2 - 6
  4. x2−x+6x^2 - x + 6
Explanation: To expand this expression, we need to apply the FOIL method. First: x × x = x². Outer: x × 2 = 2x. Inner: (-3) × x = -3x. Last: (-3) × 2 = -6. Combining all terms: x² + 2x - 3x - 6 = x² - x - 6. Choice B incorrectly has +5x instead of -x as the middle term.

Question 13

Which expression is equivalent to 2x(3+4)2x(3 + 4)?

  1. 6x+4x6x + 4x
  2. 14x14x (correct answer)
  3. 7x7x
  4. 1414
Explanation: To simplify this expression, we need to first simplify inside the parentheses, then apply the distributive property. Inside the parentheses: 3 + 4 = 7. The expression becomes 2x(7) = 14x. We can also think of this as distributing: 2x(3 + 4) = 2x(3) + 2x(4) = 6x + 8x = 14x. Choice A incorrectly leaves the expression in distributed form without combining like terms.

Question 14

Factor: x2+5x+6x^2 + 5x + 6

  1. (x+2)(x+3)(x + 2)(x + 3) (correct answer)
  2. (x+1)(x+6)(x + 1)(x + 6)
  3. (x+3)(x−2)(x + 3)(x - 2)
  4. (x−1)(x−6)(x - 1)(x - 6)
Explanation: To factor this quadratic expression, we need to find two numbers that multiply to 66 and add to 55. The numbers 22 and 33 satisfy these conditions: 2×3=62 \times 3 = 6 and 2+3=52 + 3 = 5. Therefore, x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3). We can verify by expanding: (x+2)(x+3)=x2+3x+2x+6=x2+5x+6(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6. Choice B incorrectly uses factors 11 and 66, which would give x2+7x+6x^2 + 7x + 6.

Question 15

Simplify: 4(y+3)−2y4(y + 3) - 2y

  1. 4y+124y + 12
  2. 2y+122y + 12 (correct answer)
  3. 4y+64y + 6
  4. 2y+62y + 6
Explanation: First, apply the distributive property to expand 4(y + 3), which gives 4y + 12. Then combine this with -2y to get 4y + 12 - 2y. Combine like terms by grouping the y terms: (4y - 2y) + 12 = 2y + 12. The simplified form is 2y + 12.

Question 16

Simplify: (3z+2)+(5−z)(3z + 2) + (5 - z).

  1. 2z+72z + 7 (correct answer)
  2. 2z−32z - 3
  3. 4z+74z + 7
  4. 4z+34z + 3
Explanation: Remove parentheses and combine like terms. Group the z terms: 3z - z = 2z. Group the constants: 2 + 5 = 7. Therefore, (3z + 2) + (5 - z) = 3z + 2 + 5 - z = 2z + 7. Choice C incorrectly shows 4z instead of 2z for the coefficient.

Question 17

Factor: x2+6xx^2 + 6x.

  1. x2(1+6)x^2(1 + 6)
  2. (x+3)2(x + 3)^2
  3. x(x+6)x(x + 6) (correct answer)
  4. 6(x+1)6(x + 1)
Explanation: Factor out the greatest common factor from both terms. Both x² and 6x contain the factor x: x² = x(x) and 6x = x(6). Therefore, x² + 6x = x(x + 6). Choice B represents a perfect square trinomial, which this expression is not.

Question 18

Factor: $12k - 8$.

  1. 12(k−8)12(k - 8)
  2. 2(6k−8)2(6k - 8)
  3. 4(3k+2)4(3k + 2)
  4. 4(3k−2)4(3k - 2) (correct answer)
Explanation: Find the greatest common factor of 12k and 8. Both terms share a factor of 4: 12k = 4(3k) and 8 = 4(2). Therefore, 12k - 8 = 4(3k - 2). Choice B factors out 2 instead of 4, while choice C has the wrong sign inside the parentheses.

Question 19

Which expression is equivalent to 3(x+4)−2x3(x + 4) - 2x?

  1. x+12x + 12 (correct answer)
  2. 5x+125x + 12
  3. x+4x + 4
  4. x+8x + 8
Explanation: Apply the distributive property first: 3(x + 4) = 3x + 12. Then subtract 2x from this result: 3x + 12 - 2x = x + 12. Combine like terms: 3x - 2x = x, and the constant remains 12. Choice B incorrectly keeps the 3x term without subtracting 2x.

Question 20

If S=n2(a+L)S = \frac{n}{2}(a + L), which of the following equations correctly expresses LL in terms of SS, nn, and aa?

  1. L=2S−naL = 2S - na
  2. L=2S−naL = \frac{2S - n}{a}
  3. L=S2n−aL = \frac{S}{2n} - a
  4. L=2Sn−aL = \frac{2S}{n} - a (correct answer)
Explanation: Solving literal equations (equations with multiple variables) requires the same algebraic steps as solving for x, just with more symbols. Starting with S=n2(a+L)S = \frac{n}{2}(a + L), first clear the fraction by multiplying both sides by 2: 2S=n(a+L)2S = n(a + L). Divide by n: 2Sn=a+L\frac{2S}{n} = a + L. Finally, subtract a: L=2Sn−aL = \frac{2S}{n} - a. Strategy: Work systematically and keep track of which variable you're solving for—treat all other variables as if they were numbers.