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equivalent expressions

SAT algebra is loaded with equivalent expressions: simplifying, factoring, expanding, and rearranging equations. Recognizing when two expressions mean the same thing is a secret weapon for SAT success. This page goes deep: matching forms, distributing, factoring, and handling all the SAT’s favorite “which of the following is equivalent?” questions.

what are equivalent expressions?

Two expressions are equivalent if they give the same value for every possible value of the variable. On the SAT, you’ll often be given expressions that look different but are algebraically the same. To check for equivalence, use your skills with the distributive property, factoring, and combining like terms. Sometimes, the test will also ask you to “reverse engineer” one form into another.

Example:
3(x+2)x=3x+6x=2x+63(x + 2) - x = 3x + 6 - x = 2x + 6

Here, both 3(x+2)x3(x + 2) - x and 2x+62x + 6 are equivalent—they’ll give the same answer for any xx.

To check, plug in a number (like x=1x = 1): both sides give 8.

Always distribute and combine like terms carefully—equivalent expressions might look different, but their values always match!

distributing, factoring, rearranging

The SAT will test your comfort with the distributive property, factoring, and rearranging algebraic expressions—sometimes all in one problem! Here’s how each process works, with a challenging example for each:

Distributive property (Practice):

What is the value of kk if 3(k+2)2(k4)=193(k + 2) - 2(k - 4) = 19?

Step 1: Distribute the 33 into (k+2)(k + 2) and the 2-2 into (k4)(k - 4).
Step 2: Combine all the like terms (all kk's together, all constants together).
Step 3: Solve for kk like a normal equation.

3(k+2)2(k4)=193k+62k+8=19k+14=19k=53(k + 2) - 2(k - 4) = 19 \\ 3k + 6 - 2k + 8 = 19 \\ k + 14 = 19 \\ k = 5

Use the distributive property, combine terms, and solve step by step: k = 5.

Factoring (Practice):

Factor completely: 6x215x6x^2 - 15x.

Step 1: Look for the greatest common factor (GCF) of both terms. Here, both 6x26x^2 and 15x-15x share a factor of 3x3x.
Step 2: Factor out 3x3x by dividing each term by 3x3x:

6x2÷3x=2x6x^2 \div 3x = 2x
15x÷3x=5-15x \div 3x = -5

Step 3: Write as a product: 3x(2x5)3x(2x - 5).

6x215x=3x(2x5)6x^2 - 15x = 3x(2x - 5)

Always check by distributing back: 3x(2x)=6x23x(2x) = 6x^2 and 3x(5)=15x3x(-5) = -15x.
Factoring is about "un-distributing": pull out anything both terms share.

Expanding/Combining (Practice):

Simplify completely: 4(x2)3[2x(x5)]4(x - 2) - 3[2x - (x - 5)].

Step 1: Start by expanding each grouping, being careful with negatives—especially when subtracting an entire parenthesis.
Step 2: Combine like terms: gather all the xx's, then all constants.

4(x2)3[2x(x5)]=4x83[2xx+5]=4x83(x+5)=4x83x15=x234(x - 2) - 3[2x - (x - 5)] \\ = 4x - 8 - 3[2x - x + 5] \\ = 4x - 8 - 3(x + 5) \\ = 4x - 8 - 3x - 15 \\ = x - 23

Expand, distribute, and combine step by step. Answer: x - 23.

SAT tip: Don’t rush factoring—always look for a GCF first, and double-check by redistributing to be sure! Matching the answer’s form is key for test success.

Let's practice with a few College Board questions!

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Question 1

Which of the following is equivalent to the expression
(2x+5)2(x2)+2(x+3)(2x + 5)^2 - (x - 2) + 2(x + 3)

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Question 2

Which of the following expressions is equivalent to x25x^2 - 5?

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Question 3

The equation
(ax+3)(5x2bx+4)=20x39x22x+12(a x + 3)(5x^2 - b x + 4) = 20x^3 - 9x^2 - 2x + 12
is true for all xx, where aa and bb are constants. What is the value of abab?

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equivalent expressions: sat mastery

You can now simplify, factor, expand, and recognize all forms of equivalent algebraic expressions the SAT can throw at you.

Next Up

26: nonlinear equations

solve and evaluate nonlinear equations