SAT Math Equivalent Expressions

Factoring, expanding, and simplifying — fast techniques

Equivalent expressions is the topic where the SAT tests pure algebra — no word context, no graphs. The question usually reads: "Which of the following is equivalent to $(x + 3) (x - 2)$ ?" Quick points if you know identities and factoring.

Expect 3–4 questions per section, on average easier than the test's overall difficulty. The key is memorizing a few identities — difference of squares, square of a sum, sum of cubes — and knowing how to factor trinomials. That handles 90% of this topic.

What the SAT actually tests

Expanding: $(x + a) (x + b) = x^{2} + (a + b) x + ab$
Factoring a quadratic trinomial
Identities: $(a + b)^{2}$ , $(a - b)^{2}$ , $a^{2} - b^{2}$
Simplifying rational expressions by canceling
Factoring out a common factor

Key concepts

Difference of squares

$a^{2} - b^{2} = (a - b) (a + b)$ . The most common SAT identity. Works with numbers too: $25 - 16 = (5 - 4) (5 + 4) = 9$ .

Square of a sum / difference

$(a + b)^{2} = a^{2} + 2 ab + b^{2}$ , $(a - b)^{2} = a^{2} - 2 ab + b^{2}$ . The middle term is $2 ab$ — don't drop the 2.

Factoring trinomials

$x^{2} + b x + c$ factors as $(x + p) (x + q)$ where $p + q = b$ and $p \cdot q = c$ .

Canceling in rational expressions

In $\frac{( x + 3 ) ( x - 2 )}{( x + 3 ) ( x + 5 )}$ , you can cancel $(x + 3)$ as long as $x \neq = - 3$ . The SAT occasionally tests that domain restriction.

Worked examples

Example 1

Which expression equals $(x + 4) (x - 4) - (x - 2)^{2}$ ?

Solution

Expand: $(x + 4) (x - 4) = x^{2} - 16$ . $(x - 2)^{2} = x^{2} - 4 x + 4$ . Subtract: $(x^{2} - 16) - (x^{2} - 4 x + 4) = 4 x - 20$ .

💡 Use the identities — they save arithmetic and mistakes.

Example 2

Simplify $\frac{x ^{2} - 9}{x ^{2} + 5 x + 6}$ .

Solution

Factor: numerator $= (x - 3) (x + 3)$ , denominator $= (x + 2) (x + 3)$ . Cancel: $\frac{x - 3}{x + 2}$ .

💡 Factor first, then cancel. Never cancel sums with sums — only factors.

Common pitfalls

Canceling sums: $\frac{x + 3}{x + 5}$ does NOT simplify to $\frac{3}{5}$ .
Dropping the $2 ab$ middle term in $(a + b)^{2}$ .
Writing $(a + b)^{2} = a^{2} + b^{2}$ . The cross term has to be there.
Guessing trinomial factorizations without verifying $p \cdot q = c$ and $p + q = b$ .

Exam strategy

Memorize the identities — that's the foundation. When you see $a^{2} - b^{2}$ , factor immediately. For any rational expression, factor numerator and denominator first; that's the only way to cancel. If you don't see a path, plug in a concrete number (say $x = 2$ ) and check which answer choice matches.

Frequently asked questions

What identities do I need to know?

Three core ones: $(a + b)^{2} = a^{2} + 2 ab + b^{2}$ , $(a - b)^{2} = a^{2} - 2 ab + b^{2}$ , $a^{2} - b^{2} = (a - b) (a + b)$ . Plus factoring $a x^{2} + b x + c$ .

How do I factor a quadratic?

For $x^{2} + b x + c$ : find two numbers that sum to $b$ and multiply to $c$ . Example: $x^{2} + 5 x + 6 = (x + 2) (x + 3)$ since $2 + 3 = 5$ and $2 \cdot 3 = 6$ .

How do I simplify rational expressions?

Factor numerator and denominator. Cancel common factors. Example: $\frac{x ^{2} - 4}{x - 2} = \frac{( x - 2 ) ( x + 2 )}{x - 2} = x + 2$ (for $x \neq = 2$ ).

Can I just plug in a number to check?

Yes — this is legal and effective. Pick an "odd" number (say $x = 3$ , not 0 or 1), compute the original, then check which answer choice matches. Great for "which expression is equivalent to…" questions.

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