# Equivalent Expressions on the SAT: Examples & Practice Source: https://1600.now/sat-skill/equivalent-expressions --- Math · Advanced Math ## Equivalent Expressions Equivalent expressions give the same value for every input where both are defined. Factor, expand, or complete the square to reveal the feature the question needs, while preserving restrictions. Written by [Luke Finigan](https://1600.now/about) 2 min read Updated Oct 2, 2026 ### Cancel factors, not terms For $\frac{x^2-9}{x-3}$, factor the numerator: $\frac{(x-3)(x+3)}{x-3}=x+3$ when $x\ne3$. The original expression is undefined at 3, even though the simplified expression has a value there. You cannot cancel the $x$ terms in $\frac{x+3}{x+5}$. Addition does not create a shared multiplicative factor. A numerical substitution can quickly expose that invalid step. ### Rewrite a quadratic to reveal its vertex Complete the square in $x^2-6x+11$. Half of $-6$ is $-3$; its square is 9. Thus $x^2-6x+11=(x-3)^2+2$. This shows a minimum of 2 at input 3. If the leading coefficient is not 1, factor it out of the variable terms first. For $2x^2-12x+11$, the result is $2(x-3)^2-7$. The added square inside the parentheses is also multiplied by 2. ### Apply exponent rules to the correct operation Multiplication of equal bases adds exponents: $a^3a^4=a^7$. A power of a power multiplies them: $(a^3)^4=a^{12}$. Neither rule applies to $a^3+a^4$, which is a sum. For radicals, $\sqrt{49x^2}=7|x|$ for real $x$. If the problem states $x\ge0$, it becomes $7x$. Read the stated domain before dropping an absolute value. ### Use substitutions to reject choices, then verify the algebra Testing $x=2$ can eliminate an expression that disagrees with the original at 2. Avoid inputs such as 0 or 1 when they make several distinct choices collapse to the same value. Agreement at one input does not prove two expressions equivalent. After eliminating choices, factor or expand to verify the remaining expression for the allowed domain. ### Check your understanding Try it yourself: For $x\ne4$, which expression equals $\frac{x^2-16}{x-4}$? · A $x-4$ B $x+4$ C $x^2+4$ D $x+16$ ### Sample equivalent expressions questions Equivalent Expressions · Easy: Which expression is equivalent to $72x^2$? · A $72(1+x^2)$ B $(8x)(9x)$ C $(8x^2)(9x^2)$ D $(72x^2)(x)$ - [Open the question and explanation](https://1600.now/bank/math/e2459660) Equivalent Expressions · Easy: If $6m = 24$, what is the value of $6m - 9$? · A 4 B 135 C 15 D 33 - [Open the question and explanation](https://1600.now/bank/math/aef2a838) Equivalent Expressions · Easy: Which expression is equivalent to $7x(x+4)$? · A $7x^{2}+4$ B $7x^{2}+28x$ C $8x^{2}+4$ D $8x^{2}+11x$ - [Open the question and explanation](https://1600.now/bank/math/fe131eaa) ### Practice this skill 1. Name the feature you want to expose: zeros, vertex, factor, or exponent. 2. Keep excluded inputs beside any rational simplification. 3. Explain the algebraic identity instead of relying on one numerical match. [Practice equivalent expressions](https://1600.now/bank/math/skill/Equivalent%20expressions) [Print the equivalent expressions worksheet and worked answers](https://1600.now/sat-equivalent-expressions-worksheet). Then use a [mixed practice module](https://1600.now/modules) to check whether you can choose this method among other question types. ### Source and question labels The bank label for this guide is "Equivalent expressions" in Advanced Math. The worked examples above are written for this guide. - [College Board: Types of Math tested](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types) ### Related skills - **[Nonlinear Functions](https://1600.now/sat-skill/nonlinear-functions)** The form of a nonlinear function tells you what you can read directly. - **[Nonlinear Equations and Systems](https://1600.now/sat-skill/nonlinear-equations-and-systems)** Choose a method that fits the equation: factor a quadratic, isolate a radical before squaring, split an absolute value into cases, or substitute to find graph intersections. - **[Linear Equations in One Variable](https://1600.now/sat-skill/linear-equations-one-variable)** Keep an equation balanced: simplify each side, collect the variable terms, and undo multiplication or division.