Advanced Math · Math
Equivalent expressions
Rewriting expressions in equivalent form: expanding, factoring, simplifying rational expressions, and applying exponent rules.
60 questions in the bank · 5 easier · 28 medium · 27 harder · this domain is ≈35% of the section
What to watch for
Expanding your answer back out is a fifteen-second check that catches almost every sign error on these. For exponents, keep the rules separate: powers of powers multiply, same-base products add.
How the question is worded
These are the actual stems used in the bank — the SAT reuses a small set of wordings, and recognising them saves time on test day.
- The expression (x + 3)(x − 2) is equivalent to which of the following?
- The expression (x + 6)(x + 7) is equivalent to which of the following?
- The expression (2x + 2)(x − 4) is equivalent to which of the following?
- The expression (x + 6)(x − 2) is equivalent to which of the following?
Worked examples
easyThe expression (x + 3)(x − 2) is equivalent to which of the following?
The expression (x + 3)(x − 2) is equivalent to which of the following?
- A. x² − x − 6
- B. x² + x − 6
- C. x² − 5x − 6
- D. x² + x + 6
Answer: B. x² + x − 6
Explanation
Distribute each term: (x + 3)(x − 2) = x² + −2x + 3x + −6 = x² + x − 6. The middle coefficient is 1·−2 + 3 = 1 — multiplying before adding is where the wrong answers come from.
mediumThe expression (2x + 2)(x − 4) is equivalent to which of the following?
The expression (2x + 2)(x − 4) is equivalent to which of the following?
- A. 2x² − 6x + 8
- B. 2x² − 2x − 8
- C. 2x² − 6x − 8
- D. 2x² − 10x − 8
Answer: C. 2x² − 6x − 8
Explanation
Distribute each term: (2x + 2)(x − 4) = 2x² + −8x + 2x + −8 = 2x² − 6x − 8. The middle coefficient is 2·−4 + 2 = −6 — multiplying before adding is where the wrong answers come from.
hardThe expression x² + 8x + 9 can be rewritten as (x + 4)² + k, where k is a cons…
The expression x² + 8x + 9 can be rewritten as (x + 4)² + k, where k is a constant. What is the value of k?
Answer: -7
Explanation
(x + 4)² = x² + 8x + 16. To match x² + 8x + 9, k must supply the rest of the constant: k = 9 − 16 = −7. Completing the square is exactly this bookkeeping on the constant term.