Math

Equivalent Expressions

1 min readEasy5-question drill

Equivalent expressions questions test whether you can rewrite algebra without changing its value — the core skill behind half the algebra on the test. Master a few rules and these become free points.

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Two expressions are equivalent if they give the same value for every possible input. 2(x+3) and 2x+6 are equivalent because no matter what x is, both produce the same number. The test asks you to rewrite an expression into an equivalent form — usually by expanding, factoring, or combining like terms.

Here are the tools you need:

1. Distributive property. Multiply the outside term by each inside term: a(b+c) = ab + ac. So 3(2x² + 4x - 1) = 6x² + 12x - 3.

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Distributing 3(2x² + 4x − 1)

Step	Multiply	Result
First term	3 · 2x²	6x²
Second term	3 · 4x	12x
Third term	3 · (−1)	−3

Distribute to every term: 6x² + 12x − 3.

2. FOIL (for two binomials). A binomial is a two-term expression like (x+3). FOIL multiplies First, Outer, Inner, Last: (x+3)(x-5) = x·x + x·(-5) + 3·x + 3·(-5) = x² - 5x + 3x - 15. Then combine the middle terms: x² - 2x - 15.

3. Combining like terms. Like terms have the exact same variable part. 3x and 5x are like terms (both have x); 3x and 3x² are NOT (different powers). You can only add/subtract like terms: 3x + 5x = 8x, but x² + x stays as is.

4. Exponent rules. When the bases match:

aᵐ · aⁿ = aᵐ⁺ⁿ (multiply → add exponents)
aᵐ / aⁿ = aᵐ⁻ⁿ (divide → subtract exponents)
(aᵐ)ⁿ = aᵐⁿ (power of a power → multiply)

These matter for problems like 2^(x+1) = 32. Rewrite 32 as a power of 2: 32 = 2⁵. Now both sides have base 2, so the exponents must be equal: x + 1 = 5, giving x = 4.

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Do both sides of the exponent equation have the same base?

Yes ↓

→ Set the exponents equal and solve.

No ↓

→ Rewrite one side as a power of the other base first (e.g. 32 = 2⁵).

Strategy for solving exponential equations.

Strategy: If the answer choices are expanded (no parentheses) but the question is factored, expand. If they're factored, factor. When totally stuck, you can plug in a number (say x = 2) into the original and into each choice — the equivalent one matches.

Quick check

Check your understanding with a question from this topic:

If 2^(x+1) = 32, what is the value of x?

Your answer:

Enter a whole number, fraction (e.g. 3/4), or decimal (e.g. .75).

Worked examples

Example 1

Which expression is equivalent to 4(3x² - 2x + 5)?

Example 2

Which expression is equivalent to (2x - 3)(x + 4)?

Example 3

If 3^(2x-1) = 81, what is the value of x?

Common pitfalls

Forgetting to distribute to every term

Students multiply the outside number by the first term only. 4(3x² - 2x + 5) is NOT 12x² - 2x + 5 — the 4 hits the -2x and the 5 too.

Combining unlike terms

x² and x are different terms and can never be added together. Only combine terms with the identical variable AND exponent.

Sign errors when a term is negative

A negative inside the parentheses or in a binomial flips signs. In (2x-3)(x+4), the -3 makes the Inner term -3x and the Last term -12. Track every minus sign carefully.

Not matching bases in exponent equations

You can only equate exponents when both sides have the SAME base. Rewrite numbers like 32, 81, 64 as powers (2⁵, 3⁴, 2⁶) first.

Key takeaways

Equivalent means equal for every value of the variable — verify by plugging in a number if unsure.
Distribute to EVERY term; FOIL multiplies First, Outer, Inner, Last then combines the middle terms.
Only combine like terms — same variable, same exponent.
For exponent equations, rewrite both sides with the same base, then set the exponents equal.
When stuck, plug in a value (like x=2) into the original and each choice — the match is equivalent.

Watch & learn

Curated Khan Academy walkthroughs on Equivalent Expressions. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.

Tracks your progress across lessons.

Try it yourself

5 practice questions on Equivalent Expressions, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Equivalent Expressions

You are here

→Nonlinear Equations in 1 Variable and Systems of Equations in 2 Variables →Nonlinear Functions →Linear Equations in 1 Variable