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Equivalent expressions

Rewrite polynomial, rational, and exponent expressions into an equivalent form without changing their value.

Advanced MathMath
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What the SAT actually asks

You transform an expression into an equivalent one. That means expanding products of binomials, factoring, combining rational expressions over a common denominator, simplifying with exponent rules, and converting between radical and fractional-exponent notation. The question rarely asks you to solve anything. It gives you an expression, often with a stated restriction such as x not equal to 2, and asks which choice is equivalent. Answer choices are usually close variants that differ by one sign or one coefficient.

Equivalent expressions usually make up about 3 to 4 of the 44 math questions, inside the roughly 35 percent Advanced Math share.

Where students lose points

How to attack these questions

  1. 1Choose one direction and commit: either expand everything or factor everything, rather than doing a bit of both.
  2. 2When combining fractions, write the common denominator once and work only on the numerator underneath it.
  3. 3If the algebra gets long, substitute a simple number such as x equals 3 into the original and into each choice, and keep the match.

Example questions

Written for this skill, with the answer explained. 162 more are in the app.

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Equivalent expressions
Real question

Solve for x: x² = 36

Medium

The graphs of y = 2x and y = x^2 intersect at two points. What is the nonzero x-coordinate of an intersection point?

  • A) 1
  • B) 2 ✓
  • C) 4
  • D) -2

Why: At an intersection the y-values are equal, so set x^2 = 2x. Move everything to one side and factor: x^2 - 2x = 0, so x(x - 2) = 0 and x = 0 or x = 2. The nonzero x-coordinate is 2. Answering 4 gives the y-coordinate of that intersection point, since y = 2(2) = 4. Answering -2 flips the sign, and there x^2 = 4 while 2x = -4. Answering 1 satisfies neither equation as a match, since 2x = 2 but x^2 = 1.

Medium

If a + b = 9 and a^2 + b^2 = 53, what is the value of ab?

  • A) 14 ✓
  • B) 18
  • C) 28
  • D) 34

Why: Square the sum: (a + b)^2 = a^2 + 2ab + b^2, so 81 = 53 + 2ab. Subtract 53: 2ab = 28. Divide by 2: ab = 14. Answering 28 stops at 2ab without halving it. Answering 18 doubles the sum a + b instead of working with the squares. Answering 34 is larger than the product can be here, since the two numbers are 7 and 2, and 7 + 2 = 9 with 49 + 4 = 53.

Harder

If (2x - 3)^2 = 4x^2 + bx + 9 for all values of x, what is the value of b^2?

  • A) 12
  • B) 36
  • C) 81
  • D) 144 ✓

Why: Expanding gives 4x^2 - 12x + 9, so b = -12 and b^2 = 144. Choice A reports b without squaring it, and choice B uses 6 rather than 2(2)(3) = 12 for the middle coefficient. Answering 36 means you squared 6, forgetting the factor of 2 from 2x. Answering 81 means you squared the constant 9 instead of b.

Questions students ask

Can I plug in numbers instead of doing the algebra?

Yes. Pick a value that is easy and allowed by the restrictions, such as 3 or 4, evaluate the original expression, then evaluate each answer choice. Keep the choice that matches. If two choices match, try a second number to break the tie.

Why does the question say x is not equal to some number?

Those values would make a denominator zero, so the original expression is undefined there. The restriction is there to keep the rewrite legitimate. It does not change how you simplify, but it does tell you which factor is in the denominator.

How do I simplify an expression with a fractional exponent?

A fractional exponent has the denominator as the root and the numerator as the power, so x to the two-thirds is the cube root of x squared. Convert to fractional exponents, apply the exponent rules, then convert back if the answer choices use radicals.

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Questions on equivalent expressions alone, with explanations on every one, and the app tracks which ones you keep missing.

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