The sixth most common SAT Math question type (6.1% of the bank): rewriting rather than solving. Covers the substitution test that turns algebra into arithmetic, the exponent rules, the three factoring patterns, distributing a negative across every term, cancelling factors rather than terms in rational expressions, and combining fractions over a common denominator — with six worked examples, four traps and three checks.
Subject: SAT Prep · 61 slides · applied lesson
Open the interactive version of this deck
Title
SAT Math · Type 6 of 19
6.1% of the question bank — 102 of 1675 questions
Objectives
Here you are not solving anything. You are handed an expression and asked which of four others is the same thing written differently. The mathematics is factoring, expanding and the exponent rules — but the fastest reliable method on the test is not algebra at all. It is picking a number, evaluating everything, and seeing which choice matches.
One tactic underlies the whole deck: equivalence means agreement at every input, so testing one well-chosen input is genuine evidence.
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — 102 tagged questions of this type in the site's bank
Section
Section 1
Concept
Here you are not solving anything. You are handed an expression and asked which of four others is the same thing written differently. The mathematics is factoring, expanding and the exponent rules — but the fastest reliable method on the test is not algebra at all. It is picking a number, evaluating everything, and seeing which choice matches.
You will see it phrased in these ways:
The giveaway is that there is no equals sign with an unknown to find, or if there is one, it holds for every x rather than for a particular x. Nothing here has a solution set.
Picture it
The same three-panel card as the survey deck, so the shorthand carries over: what identifies it, what you write first, and what is built to catch you.
Figure (svg): Equivalent expressions: the tell, the move, and the trap
The green panel is unusual: on most types the move is a piece of mathematics, and here it is a piece of tactics. Use it even when you can see the algebra, because it checks itself.
Prediction
Recognition matters because the method is completely different from solving.
Predict first
Three of these ask about equivalence. Which one asks you to solve?
Correct: For what value of x is 3x plus 7 equal to 22?
Why: That question has a solution — a particular x that makes the statement true — which makes it type 7, a linear equation in one variable. The other three hold for EVERY value of x, which is what equivalence means. The phrase for all x in the fourth option is the clearest possible signal, and its presence tells you immediately that substituting any convenient number is a legitimate test.
Concept
Four sub-topics wear this one label, and the substitution test works on all of them.
| sub-topic | what it looks like | the algebra behind it |
|---|---|---|
| Factoring and expanding | brackets appearing or disappearing | difference of squares, trinomials, GCF |
| Exponent rules | powers multiplied, divided, or raised | add, subtract and multiply the exponents |
| Rational expressions | a fraction with variables above and below | cancel common factors only |
| Radicals and fractional exponents | roots rewritten as powers | a fractional exponent is a root |
Factoring and exponents together are the large majority. If you are short of time, learn those two completely.
Definition probe
The SAT reuses three patterns constantly. Sort these.
Sort into buckets
Which pattern does each expression match?
Discrimination
Six rewriting steps. Which are valid for all x?
Sort into buckets
Is the step legal?
Warm-up
Do it the slow way and see how fast it actually is.
Discussion prompt
Which is equivalent to (x plus 4)(x minus 4): (A) x squared minus 16, (B) x squared plus 16, (C) x squared minus 8x minus 16, or (D) x squared minus 8?
Hint: Pick x equals 2 and evaluate everything.
Answer:
Substitute x equals 2. The original is (6)(negative 2), which is negative 12.
Now the choices: (A) 4 minus 16 is negative 12. (B) 4 plus 16 is 20. (C) 4 minus 16 minus 16 is negative 28. (D) 4 minus 8 is negative 4.
Only (A) matches, and the whole test took about fifteen seconds of arithmetic with no risk of an algebra slip.
The algebra confirms it: this is a difference of two squares, so it expands to x squared minus 16. But you did not need to know that pattern in order to answer correctly.
Pattern
Three steps, and the middle one is arithmetic rather than algebra.
Choose a number for the variable: small, and not 0 or 1.
Why: Zero and one are too well behaved — they make different expressions agree by accident. Two or three usually separates every choice.
Evaluate the original expression and every answer choice at that number.
Why: Equivalence means agreement for all inputs, so any disagreement at one input disproves it immediately.
If two choices still match, test a second number and discard the one that now fails.
Why: Ties are rare and always broken by a second value. Choosing 5 or a negative usually does it.
Use algebra when the expression is simple enough to rewrite in one line. Use substitution the rest of the time, and especially when you feel unsure.
Section
Section 2
Concept
Two expressions are equivalent when they produce the same value for every value of the variable that both allow.
In practice on a four-choice question, one well-chosen number nearly always leaves exactly one survivor.
Concept
Multiplying adds exponents, dividing subtracts them, and a power of a power multiplies them.
| expression | becomes | why |
|---|---|---|
| x to the a times x to the b | x to the (a plus b) | the factors are counted together |
| x to the a over x to the b | x to the (a minus b) | the shared factors cancel |
| (x to the a) to the b | x to the (a times b) | a groups of b factors each |
| x to the 0 | 1 | any nonzero base to the power zero |
| x to the negative a | 1 over x to the a | a negative exponent means a reciprocal |
| x to the one half | the square root of x | a fractional exponent is a root |
If you can never remember which is which, expand a tiny case: x squared times x cubed written out is five x's, so the exponents add.
Prediction
The most confused pair of rules.
Predict first
Which is equivalent to (x to the fourth) times (x to the third)?
Correct: x to the seventh
Why: Multiplying powers of the same base adds the exponents: 4 plus 3 is 7. Writing it out makes it obvious — four x's times three x's is seven x's. The answer x to the twelfth multiplies the exponents, which is the rule for a power raised to a power. The 2x version invents a coefficient that was never there.
Concept
Pull out a common factor first, then check for a difference of squares or a trinomial.
| pattern | form | factors as |
|---|---|---|
| Common factor | 6x cubed plus 9x squared | 3x squared (2x plus 3) |
| Difference of squares | a squared minus b squared | (a plus b)(a minus b) |
| Trinomial | x squared plus bx plus c | two numbers multiplying to c and adding to b |
| Perfect square | x squared plus 2ax plus a squared | (x plus a) squared |
Always try the common factor first. It is the cheapest step and it frequently reveals one of the other patterns underneath.
Prediction
Across every term.
Predict first
Which is equivalent to 7x minus (3x minus 5)?
Correct: 4x plus 5
Why: The minus sign applies to the whole bracket, so it becomes 7x minus 3x plus 5, which is 4x plus 5. Substituting x equals 2 confirms it: the original is 14 minus 1, which is 13, and 4 times 2 plus 5 is also 13. The second choice, 4x minus 5, gives 3 and is the answer you get by distributing to only the first term.
Concept
Subtracting a bracket changes the sign of every term inside it, not just the first.
This single error accounts for a large share of the wrong answers on this type, and it is completely invisible unless you substitute.
Concept
In a fraction you may cancel something multiplied through the whole numerator, never something merely added to part of it.
Test any cancellation by substituting. Partial cancelling is the single most common algebra error that survives from school into calculus.
Prediction
Factors cancel; terms do not.
Predict first
Which is equivalent to (3x squared plus 12x) over 3x, for x not zero?
Correct: x plus 4
Why: Factor the numerator first: 3x(x plus 4). The 3x cancels with the denominator as a whole factor, leaving x plus 4. Substituting x equals 2 checks it: the original is (12 plus 24) over 6, which is 6, and 2 plus 4 is 6. Cancelling only part of the numerator gives the other choices, all of which fail at x equals 2.
Concept
Fractions add only over the same denominator, and the numerators are added after each is scaled.
Substituting x equals 2 and y equals 3 settles any doubt instantly: the true value is five sixths, and the wrong version gives two fifths.
Concept
x to the power one over n is the nth root of x, and x to the power m over n is the nth root of x to the m.
A memory hook that survives pressure: the root is on the bottom, the same way a denominator is.
Prediction
Squaring a bracket is not squaring its terms.
Predict first
Which is equivalent to (x plus 5) squared?
Correct: x squared plus 10x plus 25
Why: Squaring means multiplying the bracket by itself, which produces four products: x times x, x times 5 twice, and 5 times 5. The two middle products combine into the cross term 10x. Substituting x equals 1 separates them immediately: the original is 36, and only the first choice gives 1 plus 10 plus 25, which is 36.
Two truths and a lie
Three of these rewriting rules are correct. The one left standing is false.
Eliminate the wrong options
Which statement is FALSE?
Survives elimination: b
Why: Squaring a sum produces a cross term: (x plus y) squared is x squared plus 2xy plus y squared. Dropping the 2xy is probably the single most common algebra error in existence, and it is easy to disprove — take x and y both equal to 1, and the left side is 4 while the claimed right side is 2. Note the contrast with statement C, where the cross terms genuinely do cancel, because one bracket has a plus and the other a minus.
Check
Look for the pattern before doing anything else.
Check your understanding
Which expression is equivalent to 16x squared minus 81?
Answer: A
Why: Both terms are perfect squares: 16x squared is (4x) squared and 81 is 9 squared, separated by a minus. So the difference-of-squares pattern gives (4x plus 9)(4x minus 9). Checking at x equals 1: the original is 16 minus 81, which is negative 65, and (13)(negative 5) is also negative 65.
Substituting x equals 1 separated all four choices here, which is unusual — normally 1 is too weak. It worked because the choices differ in their constants as well as their coefficients.
Section
Section 3
Worked example
Which expression is equivalent to 9x squared minus 49?
Figure (svg): An area model showing the square of 3x plus 7 with its two cross rectangles
Recognise both terms as perfect squares: 9x squared is (3x) squared and 49 is 7 squared.
Why: The pattern applies whenever both terms are squares and separated by a minus.
Apply the pattern: a squared minus b squared equals (a plus b)(a minus b), with a as 3x and b as 7.
Why: Substituting into the identity is the whole step.
So it factors as (3x plus 7)(3x minus 7).
Why: The two cross terms, plus 21x and minus 21x, cancel — which is why no middle term appears.
The squares are frequently disguised by coefficients. Look for a perfect square in front of the variable as well as at the end.
Verify: check at x equals 2: the original is 36 minus 49, which is negative 13, and (13)(negative 1) is also negative 13.
Why: Agreement at a test value confirms the factoring without re-expanding.
Answer: (3x + 7)(3x - 7)
Worked example
Which expression is equivalent to 8x minus 3(2x minus 5)?
Figure (svg): Bars comparing the correct and incorrect results at x equals five
Distribute the negative 3 across BOTH terms: negative 3 times 2x is negative 6x, and negative 3 times negative 5 is plus 15.
Why: A negative multiplied by a negative gives a positive — this is the term that gets lost.
So the expression is 8x minus 6x plus 15.
Why: Both products are written before any collecting.
Collect: 2x plus 15.
Why: Only the x-terms combine; the constant stands alone.
Substituting a single number separates the right answer from the sign-error answer instantly, and the sign-error answer is always among the choices.
Verify: check at x equals 5: the original is 40 minus 3 times 5, which is 40 minus 15, or 25, and 2 times 5 plus 15 is 25.
Why: The values agree, whereas the wrong version 2x minus 15 would give negative 5.
Answer: 2x + 15
Worked example
Which expression is equivalent to (2x cubed) to the power 4, divided by (4x to the fifth)?
Figure (svg): A table tracking the coefficient and the power of x through each step
Raise the whole bracket: (2x cubed) to the 4 means 2 to the 4 times x to the 3 times 4, giving 16x to the twelfth.
Why: The exponent outside applies to the coefficient as well as to the variable.
Now divide: 16 over 4 is 4, and x to the twelfth over x to the fifth is x to the seventh.
Why: Coefficients divide normally; exponents of the same base subtract.
So the result is 4x to the seventh.
Why: Coefficient and power recombined.
The verify step names its own limitation: x equals 1 is a weak test because it hides every exponent. Use it for the coefficient and a second value for the power.
Verify: check at x equals 1: the original is 16 over 4, which is 4, and 4 times 1 is 4.
Why: x equals 1 checks only the coefficient, so also test x equals 2 for the exponent.
Answer: 4x to the seventh
Step zero
The tactic only works if the number is well chosen.
Discussion prompt
You decide to test an equivalence question by substituting. Why are 0 and 1 poor choices, and what should you use instead?
Hint: What do 0 and 1 do to powers and products?
Answer:
x equals 1 hides every exponent. One squared, one cubed and one to the tenth are all 1, so expressions differing only in a power will agree.
x equals 0 kills every term containing x, so expressions differing anywhere except the constant will agree.
Both therefore produce false matches, leaving two or three choices still standing.
Use 2 or 3. They are small enough for mental arithmetic and large enough that powers, coefficients and signs all separate.
Avoid any value that makes a denominator zero, since that input is not legal for the original expression.
If two choices still tie, test a second number — a negative one such as negative 2 is particularly good at exposing sign errors.
Worked example
Which expression is equivalent to (x squared minus 9) over (x squared plus 7x plus 12), for x not equal to negative 3 or negative 4?
Figure (svg): A number line marking the two excluded values
Factor the numerator as a difference of squares: (x plus 3)(x minus 3).
Why: Both terms are perfect squares separated by a minus.
Factor the denominator: two numbers multiplying to 12 and adding to 7 are 3 and 4, giving (x plus 3)(x plus 4).
Why: The standard trinomial search.
The factor (x plus 3) appears in both, so it cancels, leaving (x minus 3) over (x plus 4).
Why: Cancelling is legal here because (x plus 3) is a whole factor of both, not merely a term.
Factor both parts completely before cancelling anything. Cancelling before factoring is how partial cancellation happens.
Verify: check at x equals 1: the original is (1 minus 9) over (1 plus 7 plus 12), which is negative 8 over 20, or negative two fifths, and (1 minus 3) over (1 plus 4) is negative 2 over 5.
Why: The two agree, confirming the cancellation was of a genuine common factor.
Answer: (x - 3) over (x + 4)
Worked example
Which single fraction equals 3 over x plus 2 over (x plus 1)?
Figure (svg): A panel showing the two scaled numerators over the common denominator
The common denominator is x times (x plus 1).
Why: The product of the two denominators always works, and here they share no factor so it is also the least.
Scale each fraction: 3 over x becomes 3(x plus 1) over x(x plus 1), and 2 over (x plus 1) becomes 2x over x(x plus 1).
Why: Each numerator is multiplied by whatever its denominator was missing.
Add the numerators: 3x plus 3 plus 2x, which is 5x plus 3.
Why: Only the numerators add; the denominator is already shared.
The wrong answer to watch for is 5 over (2x plus 1) — adding numerators and denominators straight across, which is never legal.
Verify: check at x equals 1: the original is 3 plus 1, which is 4, and (5 plus 3) over (1 times 2) is 8 over 2, which is 4.
Why: The values agree, confirming the scaling was applied to both numerators correctly.
Answer: (5x + 3) / (x(x + 1))
Faded example
From memory. Four lines that cover most exponent questions.
Fill in the blanks
Multiplying powers of the same base means you add the exponents. Dividing means you subtract them. Raising a power to a power means you multiply them. And a negative exponent means take the reciprocal.
Why: The first and third are the pair that get swapped under pressure. The safest recovery is to expand a tiny case: x squared times x cubed written out is five x's, so multiplication adds — and if multiplication adds, then a power of a power must be the one that multiplies.
Worked example
Which expression is equivalent to (x minus 2)(x plus 5) minus (x minus 3)(x plus 4)?
Figure (svg): A table evaluating both products at x equals two and taking their difference
Choose x equals 2 — small, and not 0 or 1.
Why: Two is usually enough to separate all four choices without awkward arithmetic.
Evaluate: the first product is 0 times 7, which is 0; the second is negative 1 times 6, which is negative 6.
Why: Each bracket is evaluated before multiplying, which keeps the arithmetic trivial.
The difference is 0 minus negative 6, which is 6. Now test each choice at x equals 2 and keep the one giving 6.
Why: Only the equivalent expression can match at this input.
The algebra took three lines and two chances to slip a sign. The substitution took one line of arithmetic. On the test, take the second route.
Verify: check by expanding: the first is x squared plus 3x minus 10, the second x squared plus x minus 12, and the difference is 2x plus 2.
Why: At x equals 2 that gives 6, matching the substitution — so both methods agree.
Answer: 2x + 2
Fill the middle
The three patterns the SAT reuses.
Fill in the blanks
x squared minus 36 factors as (x + 6)(x - 6). x squared plus 8x plus 15 factors as (x + 3)(x + 5). And 5x squared plus 15x factors as 5x(x + 3).
Why: The third is the one students skip. Checking for a common factor costs two seconds and often converts an expression that looks unfactorable into one of the standard patterns — and on the SAT the answer choices are usually written in fully factored form.
Estimation
Sometimes a rough size is enough.
Predict first
At x equals 10, roughly what is (x squared plus 3x) over x?
Correct: about 13
Why: The expression simplifies to x plus 3, which at x equals 10 is 13. Even without simplifying, the numerator is about 130 and dividing by 10 gives about 13. The answer 103 comes from cancelling only the first term, giving x squared plus 3 — the partial-cancellation error, which at x equals 10 is 103 and is therefore easy to spot as wrong once you have a rough expectation.
Check
Handle the coefficient and the power separately.
Check your understanding
Which expression is equivalent to (3x squared) to the power 3, divided by (9x to the fourth)?
Answer: A
Why: Raising the bracket gives 3 cubed times x to the 2 times 3, which is 27x to the sixth. Dividing by 9x to the fourth gives 27 over 9, which is 3, and x to the sixth over x to the fourth, which is x squared. So the answer is 3x squared. Checking at x equals 2: the original is (12) cubed over (9 times 16), which is 1728 over 144, or 12, and 3 times 4 is 12.
The x equals 2 check is the reliable one here because it tests the coefficient and the exponent at once. Testing x equals 1 would have left A and C indistinguishable from nothing.
Section
Section 4
Trap
The trap. You are given (x squared plus 6x) over x and you cancel the x in the denominator against the x in x squared, writing x plus 6x.
Or worse, you are given (x squared plus 6) over x and cancel anyway, writing x plus 6.
In the second case nothing cancels at all: the 6 has no factor of x in it, so the denominator does not divide it.
The fix. Cancelling is division, and division must be applied to the whole numerator or to none of it.
Factor the numerator first. If the denominator appears as a complete factor, it cancels; otherwise it stays.
Trap
The trap. You write (x plus 5) squared as x squared plus 25.
Squaring a bracket means multiplying it by itself, which produces four products, and the two middle ones combine into 10x.
The correct expansion is x squared plus 10x plus 25, and the error is off by 10x for every value of x except 0.
The fix. Never square terms individually. Write the bracket out twice and multiply properly, or use the pattern.
The pattern to memorise: first squared, plus twice the product, plus last squared.
Error analysis
A student's work. One step is illegal, and the substitution check catches it in seconds.
Annotate
On: \( \frac{x^2 + 6x}{x} \;=\; x + 6x \;=\; 7x \)
Partial cancelling survives into calculus if nobody names it. Name it now, and check it with a number every time you are unsure.
Trap
The trap. You write 5x minus (2x minus 7) as 5x minus 2x minus 7.
The minus applies to the whole bracket, so the second term's sign must flip too: 5x minus 2x plus 7.
The answers differ by 14, and both are among the choices.
The fix. Treat a leading minus as multiplication by negative 1 and distribute it explicitly.
Write the intermediate line with both signs changed before collecting anything.
Elimination
Which expression is equivalent to (x plus 3)(x minus 3) plus 9? Test x equals 2: the original is (5)(negative 1) plus 9, which is 4.
Eliminate the wrong options
Which choice also gives 4 at x equals 2?
Survives elimination: a
Why: The original is a difference of squares plus 9: x squared minus 9 plus 9, which is simply x squared. At x equals 2 that gives 4, matching. The other three fail at a single substitution, which is the whole point — one number and about twenty seconds of arithmetic eliminated three choices with no algebra and no risk of a sign error.
Trap
The trap. You write 1 over x plus 1 over y as 2 over (x plus y).
Fractions do not add by adding numerators and denominators. That operation has no basis at all.
The correct result is (y plus x) over xy, and substituting x equals 2 and y equals 3 gives five sixths against the wrong version's two fifths.
The fix. Scale both fractions to a common denominator, then add only the numerators.
The product of the denominators always works as a common denominator.
Counterexample
A rule students invent by analogy.
Discussion prompt
A student says: the square root of (a plus b) equals the square root of a plus the square root of b. Give a counterexample, and say what rule they are confusing it with.
Hint: Try two perfect squares.
Answer:
Counterexample: a equals 9 and b equals 16. The left side is the square root of 25, which is 5. The right side is 3 plus 4, which is 7.
So the claim fails, and it fails badly — 5 against 7.
What they are confusing it with: roots DO distribute over multiplication. The square root of (9 times 16) is the square root of 144, which is 12, and 3 times 4 is also 12.
The general pattern: roots and powers distribute over products and quotients, never over sums and differences.
The same false analogy produces (x plus y) squared equals x squared plus y squared, which is the missing-cross-term trap. It is the same mistake in the opposite direction.
Edge cases
Substituting is a proof of non-equivalence. Is it a proof of equivalence?
Discussion prompt
If a choice matches the original at x equals 2, is it definitely equivalent? Where does the test break, and how do you handle it?
Hint: How many points can two different curves share?
Answer:
No — matching at one point is not a proof. Two genuinely different expressions can agree at a particular value by coincidence.
For example x squared and 2x are different, yet both give 4 at x equals 2.
On a four-choice question this rarely matters, because the distractors are designed to differ from the original in structure, so one well-chosen value usually separates all of them.
When two choices survive, test a second value. A negative number such as negative 2 is especially good, because it exposes sign errors that a positive test misses.
The genuinely safe rule: substitution proves a choice WRONG conclusively, and makes a choice very likely right. Two agreeing values on a four-choice question is as much certainty as the format can offer.
Also avoid illegal inputs: if the original has a denominator, do not choose a value that makes it zero, because the original is undefined there and the comparison is meaningless.
Check
Two of the traps at once.
Check your understanding
Which expression is equivalent to (6x squared minus 9x) over 3x, minus (x minus 4), for x not zero?
Answer: A
Why: Factor the first part: 3x(2x minus 3) over 3x cancels to 2x minus 3. Then subtract the bracket, distributing the minus: 2x minus 3 minus x plus 4, which collects to x plus 1. Checking at x equals 2: the original is (24 minus 18) over 6, which is 1, minus (2 minus 4), which is negative 2, so 1 plus 2 is 3 — and x plus 1 at x equals 2 is 3.
The substitution check does all the work here. Three choices fail at x equals 2, and you never need to trust your own algebra.
Section
Section 5
Matching
Six rewritings. Substitute if you are unsure.
Match the pairs
Why: Note the two exponent rows sitting side by side. A power raised to a power multiplies the exponents, giving fifteen — whereas multiplying two powers would have added them to give eight. Those two rules are adjacent in the mind and constantly swapped, so seeing them next to each other is worth the space.
Sorting
Each is a rewriting a student actually made.
Sort into buckets
Is the step valid for all legal x?
Items (c) and (d) are the pair worth holding onto: roots distribute over multiplication and never over addition. Everything else on this slide follows from taking that distinction seriously.
Comparison
Fill the blanks from memory. Both methods are valid; they cost different amounts.
Comparison matrix
| method | when it wins | what it costs |
|---|---|---|
| Substitute a number | almost always on four-choice questions | arithmetic on five expressions |
| Do the algebra | when the rewrite is one obvious line | a sign error ends the question silently |
| Both | when the answer matters and time allows | twice the time, near-certainty |
| Guess by appearance | never | the distractors are built to look right |
The last row is not a joke. Every wrong choice on this type is the result of one specific plausible error, so the choice that looks most reasonable is frequently the one designed for you.
Trade off
Fill in what each candidate is good and bad for.
Comparison matrix
| test value | good because | bad because |
|---|---|---|
| x = 0 | arithmetic is trivial | kills every term containing x |
| x = 1 | arithmetic is trivial | hides every exponent |
| x = 2 | separates powers, signs and coefficients | nothing much — it is the default |
| x = -2 | exposes sign errors a positive value hides | slightly more error-prone by hand |
Start with 2. If two choices survive, go to negative 2 rather than to 3 — sign errors are far more common among SAT distractors than magnitude errors.
Real world
One minute on why rewriting is a skill at all.
Discussion prompt
If two expressions mean the same thing, why does it matter which form you write? Where does choosing a form actually change something?
Answer:
Because different forms answer different questions. That is the whole lesson of type 1: factored form shows the zeros, vertex form shows the turning point, and neither is more correct than the other.
In computing, form changes cost. Evaluating x squared plus 5x plus 6 takes more operations than evaluating (x plus 2)(x plus 3), and on millions of inputs that difference is real.
In science, form shows structure. Rewriting a formula to isolate a quantity is how you see what depends on what.
And in checking, form reveals errors. An expression that factors neatly is usually one you have written correctly; one that refuses to factor is worth re-reading.
So equivalence questions are not busywork. They are asking whether you can move between representations, which is the same skill the rest of the Math section keeps rewarding.
Ranking
All four are different expressions. Order them from smallest to largest at x equals 3.
Put in order
Why: At x equals 3: (a) is 9 minus 9, which is 0. (d) is 27 minus 20, which is 7. (b) is 6 plus 1, which is 7 — a tie with (d), so either order between them is defensible. (c) is 4 squared, which is 16. The tie is the instructive part: two different expressions agreeing at a single value is exactly the coincidence that makes one substitution suggestive rather than conclusive, and it is why a second test value breaks ties.
Concept
This type is 6.1 per cent of the section and is the most tactic-driven on the test. One session on the tactic is worth three on the algebra.
| session | what you do | why |
|---|---|---|
| 1 | Twenty equivalence questions answered ONLY by substitution, no algebra at all. | Builds trust in the tactic, which is the thing students refuse to use under pressure. |
| 2 | The exponent rules: twenty questions, saying which rule applies before computing. | Exponents are the sub-topic where the rules are adjacent and get swapped. |
| 3 | Fifteen factoring questions, checking for a common factor first every time. | The GCF step is the one most often skipped, and it simplifies everything after it. |
| 4 | Mixed set under time, with substitution as the default and algebra only as a check. | Establishes the habit you actually want on test day. |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — filter the bank to this skill tag — 102 questions, roughly a third of each difficulty
Explain it to yourself
Close the deck.
Discussion prompt
Without looking, explain why substituting a number is a valid method here, why 0 and 1 are bad choices, and what to do if two choices survive.
Hint: Start from what the word equivalent actually means.
Answer:
Why it is valid: equivalent means the two expressions agree for EVERY legal input. So a single disagreement proves non-equivalence outright.
Why not 0: it destroys every term containing x, so expressions differing anywhere but the constant will match by accident.
Why not 1: it makes every power equal to 1, so expressions differing only in exponents will match by accident.
Use 2 or 3, avoiding any value that makes a denominator zero.
If two survive, test a second value, ideally a negative one, because sign errors are the most common way distractors are built.
If you produced the reason as well as the rule, you will actually use it under pressure — which is the point.
Explain it
Two minutes, out loud.
Discussion prompt
A friend writes (x squared plus 6) over x as x plus 6. How do you show them it is wrong in a way that sticks?
Answer:
Do not argue about rules — substitute. Take x equals 2. The original is (4 plus 6) over 2, which is 5. Their answer gives 2 plus 6, which is 8.
Five is not eight, so the step is illegal. That settles it without any appeal to authority.
Then explain the mechanism: cancelling is dividing, and dividing has to hit the whole numerator. The 6 has no x in it, so the x underneath cannot divide it.
Show the contrast: (x squared plus 6x) over x IS x plus 6, because now both terms contain x. At x equals 2 that is 16 over 2, which is 8 — and 2 plus 6 is 8. It works.
Give them the habit: factor the top first. If the bottom appears as a whole factor, it cancels; if it does not, nothing does.
Commit first
Commit before you check.
Predict first
Which is equivalent to (2x minus 3) squared?
Correct: 4x squared minus 12x plus 9
Why: First squared is 4x squared, last squared is 9, and the middle term is twice the product of 2x and negative 3, which is negative 12x. Substituting x equals 1 confirms it: the original is (negative 1) squared, which is 1, and 4 minus 12 plus 9 is also 1. The second choice drops the cross term, the third halves it, and the fourth fails to square the coefficient.
Connect it up
Blank paper.
Draw it
Draw a map of equivalent expressions. Put SUBSTITUTE A NUMBER in a box at the centre, with the rule beside it: use 2, never 0 or 1, and use negative 2 to break ties. Around it, draw four branches: exponent rules with the add, subtract and multiply cases; factoring with the common factor, difference of squares and trinomial patterns; fractions with cancel-factors-not-terms and the common denominator; and signs with the distribute-across-everything rule. On each branch, write the one error that branch is famous for. Finally, in a corner, write the two false analogies: squaring a sum is not the sum of squares, and a root of a sum is not the sum of roots.
Exit ticket
One question before you close the deck.
Predict first
You face an equivalence question with four similar-looking choices and you are not certain of the algebra. What do you do?
Correct: Substitute x equals 2 into the original and all four choices
Why: Substitution converts an algebra question you are unsure about into arithmetic you are sure about, and a single disagreement disproves a choice outright. Expanding is valid but is exactly where the sign errors you are worried about would occur. Choosing by appearance is what the distractors are designed to exploit. And x equals 1 makes every power equal 1, so expressions differing only in their exponents would all match, leaving you no better off.
Recap
One type, one tactic: stop manipulating and start substituting.
| never do this | do this instead |
|---|---|
| Cancel an x against only part of the numerator | Factor the top first, then cancel whole factors |
| Write (x + 5) squared as x squared + 25 | Include the cross term: x squared + 10x + 25 |
| Distribute a minus to the first term only | Change the sign of every term inside |
| Add fractions straight across | Scale to a common denominator, add numerators only |
| Test with x = 1 | Test with x = 2, which exposes exponents |
| Trust the choice that looks tidiest | Substitute — the tidy one is often the trap |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — and the site's SAT pages to drill this type in isolation, then mixed
Want this taught 1-on-1? Alexander tutors SAT Prep — $55/session, free consultation.