Solving Equivalent Expressions Questions on the SAT

The method: Expand, Factor, or Test

Two expressions are equivalent when they produce the same value for every valid input. On the SAT you prove that by algebra (expand or factor until both sides match) or, on multiple choice, by testing smart numbers and then confirming.

Use Expand, Factor, or Test. If one side is factored and the other isn’t, expand the factored form (or factor the expanded form). If you’re matching choices, expand your expression once and compare. If algebra is messy, plug in a convenient value of x (avoid 0 and 1 when those make every choice look the same) to eliminate options, then verify the survivor.

Go faster without getting sloppy

Memorize the patterns you see constantly: (a+b)² = a² + 2ab + b², (a−b)² = a² − 2ab + b², (a+b)(a−b) = a² − b². Factoring out a GCF first often reveals one of those. For rationals, factor numerator and denominator and cancel common factors — don’t cancel terms across a plus sign.

Spot the traps in the choices

Wrong choices often expand with a sign error on the middle term, factor incompletely, or cancel illegally in a fraction. Another trap: two expressions that match at x = 2 but aren’t identical — always prefer an algebraic match when you can.

When structure hides the match

Sometimes the SAT writes an expression to look different on purpose: 2(x − 4) + 3(x − 4) is 5(x − 4) by combining like groups. Factor out the common binomial before expanding everything.

Practice until it’s automatic

Practice questions

Here are five practice problems that help you hone your skills. Use the same method we learned earlier above to solve these problems. Remember, practice makes perfect.