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SAT · Math · Advanced Math
Equivalent expressions is part of Advanced Math in the SAT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.
Question types you will see here include expand and combine like terms, factor a quadratic expression, simplify a rational expression and radicals and rational exponents.
Common traps: sign errors when distributing negatives; missing the middle term in (a + b)²; combining unlike terms.
Question 1
The expression x² - 11x + 28 can be written as (x - p)(x - q), where p < q. What is the value of q?
Enter your own answer.
Answer: 7
Find two numbers with product 28 and sum 11: 4 and 7. So x² - 11x + 28 = (x - 4)(x - 7), with p = 4 and q = 7, the larger value.
Question 2
The expression (3x - 2)(4x - 2) is equivalent to ax² + bx + c, where a, b, and c are constants. What is the value of b?
-14
-4
2
12
Answer: A. -14
Expand: (3x)(4x) + (3x)(-2) + (-2)(4x) + (-2)(-2) = 12x² + (-6 + -8)x + 4. So a = 12, b = -14, c = 4; the value of b is -14.
Question 3
A surf school's rental surcharge, in dollars, is modelled by 36x⁴ − 49. Which of the following is equivalent to this expression?
(6x² − 7)²
(6x − 7)(6x + 7)
(6x² + 7)²
(6x² − 7)(6x² + 7)
Answer: D. (6x² − 7)(6x² + 7)
36x⁴ = (6x²)² and 49 = 7², so this is a difference of squares: (6x² − 7)(6x² + 7). A and C are perfect squares, which expand with a middle term (±84x²). B treats x⁴ as x², and expands to 36x² − 49, not 36x⁴ − 49.
Question 4
The number of fragments in a coral nursery after t years is modelled by N(t) = 2t² − 12t + 22. According to the model, what is the minimum number of fragments?
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Answer: 4
Complete the square: 2t² − 12t + 22 = 2(t − 3)² − 18 + 22 = 2(t − 3)² + 4. The squared term is never negative, so the minimum value is 4 (reached at t = 3).
Question 5
Which of the following is equivalent to (2x² + 5x − 3)/(x + 3) − (x² − 1)/(x + 1) for x > 0?
x − 2
3x − 2
(2x − 1)/(x − 1)
x
Answer: D. x
2x² + 5x − 3 = (2x − 1)(x + 3) and x² − 1 = (x − 1)(x + 1), so the expression is (2x − 1) − (x − 1) = x (D). A forgets to distribute the subtraction, writing 2x − 1 − x − 1; B adds the two quotients; C divides the quotients instead of subtracting them.
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