Educerie · SAT · Math
Advanced Math · ADV.1 Equivalent expressions
What you must be able to do
| You must be able to | What it looks like on the test |
|---|---|
| Factor the four standard patterns | Which expression is equivalent to x² − 49? |
| Expand and collect | (2x − 3)(x + 5) is equivalent to… |
| Simplify a rational expression | Cancel factors, never terms |
| Use exponent and radical rules | x^(2/3) is equivalent to… |
| Complete the square | To reach vertex form, or to find a minimum |
1The idea in one paragraph
Two expressions are equivalent when they give the same value for every input. Nothing is being solved here; something is being rewritten so that a different fact becomes visible. Factored form shows you the zeros, vertex form shows you the turning point, expanded form shows you the y-intercept. The test chooses whichever form hides what it wants you to find, and the work is moving to the form that reveals it.
When four answer choices are all expressions, pick a number — say x = 2 — and evaluate the question and each option. The one that matches is the answer. It is slower than factoring but it never fails.
2The four factoring patterns
| Pattern | Recognise it by | Example |
|---|---|---|
| Common factor | Every term shares something | 6x² + 9x = 3x(2x + 3) |
| Difference of squares | Two squares, a minus between | x² − 49 = (x + 7)(x − 7) |
| Trinomial | x² + bx + c | x² + 7x + 12 = (x + 3)(x + 4) |
| Perfect square | First and last are squares, middle is twice their roots | x² − 10x + 25 = (x − 5)² |
For the trinomial, find two numbers that multiply to c and add to b. With a leading coefficient:
Always take out the common factor first. It makes everything after it easier.
3Rational expressions: cancel factors, never terms
That is legal because (x + 3) is a factor of the top. What is never legal is cancelling across a sum: (x + 5)/5 is not x, and (x² + 4)/4 is not x² + 1.
If you cannot factor the numerator, you cannot cancel anything.
4Exponents and radicals
| Rule | Example |
|---|---|
| xᵃ · xᵇ = xᵃ⁺ᵇ | x³ · x⁵ = x⁸ |
| xᵃ / xᵇ = xᵃ⁻ᵇ | x⁷ / x² = x⁵ |
| (xᵃ)ᵇ = xᵃᵇ | (x²)⁴ = x⁸ |
| x⁻ᵃ = 1/xᵃ | x⁻³ = 1/x³ |
| x^(1/n) = ⁿ√x | x^(1/2) = √x |
| x^(m/n) = ⁿ√(xᵐ) | x^(2/3) = ³√(x²) |
The fractional exponent is the one the test likes: the denominator is the root, the numerator is the power. Say it as "cube root of x squared" and it stops being frightening.
5Completing the square
Note the sign flip on the x-coordinate. This is how a question asking for a minimum value is answered when no graph is given, and it is the only reliable route when the quadratic does not factor.
6Choosing a form on purpose
| You need | Use |
|---|---|
| The zeros / x-intercepts | Factored form |
| The vertex, maximum or minimum | Vertex form |
| The y-intercept | Standard form — it is c |
A question that asks "which form displays the minimum as a constant" is asking, in the test's own language, which option is in vertex form.
Where points are lost
- Cancelling terms instead of factors in a fraction.
- Losing a sign when expanding a bracket that follows a minus.
- Misreading (x − h)² + k — the vertex is at +h, not −h.
- Forgetting the common factor first, then failing to factor what is left.
- Inverting a fractional exponent: x^(2/3) is the cube root of x squared, not the square root of x cubed.
- Adding without subtracting when completing the square, which changes the expression.
Work it right
- Take out any common factor.
- Name the pattern: two squares, a trinomial, a perfect square.
- Factor it, then check by expanding in your head.
- If nothing factors and you need a turning point, complete the square.
- If all else fails, substitute a number into the question and every option.
Try it
Q1. Which expression is equivalent to 4x² − 25?
A) (2x − 5)² B) (2x + 5)(2x − 5) C) (4x + 5)(x − 5) D) (2x − 5)(x + 5)
Q2. Which expression is equivalent to (x² + 5x − 24)/(x + 8)?
A) x − 3 B) x + 3 C) x − 8 D) x² − 3
Q3. The expression x² − 12x + 7 can be written as (x − h)² + k. What is the value of k?
A) −29 B) −5 C) 7 D) 43
Q4. Which of the following is equivalent to 2x² + 11x + 12?
A) (2x + 3)(x + 4) B) (2x + 4)(x + 3) C) (2x + 12)(x + 1) D) (x + 3)(x + 8)
Q5. If x > 0, which expression is equivalent to x^(3/4)?
A) ⁴√(x³) B) ³√(x⁴) C) 4√x / 3 D) x³ · x⁴
In one breath
Nothing here is being solved — it is being rewritten so a different fact shows. Take out the common factor first, then name the pattern: two squares with a minus, a trinomial, a perfect square. Cancel only factors, never terms across a sum, and remember that a fractional exponent puts the root underneath and the power on top. When a question wants a maximum or minimum, complete the square and read the vertex — watching the sign, because (x − h)² + k has its vertex at +h.
Answers
Q1. B. difference of squares, with 4x² = (2x)² (2x + 5)(2x − 5). A is a perfect square and expands with a middle term. C and D expand to the wrong middle terms.
Q2. A — x − 3. factor the numerator, then cancel the factor
B flips the sign. C cancels the wrong factor. D cancels a term rather than a factor.
Q3. A — −29. halve 12, square it, add and subtract
C is the original constant; D adds 36 rather than subtracting.
Q4. A. split the middle term using 2 × 12 = 24
Expanding B gives 2x² + 10x + 12 and C gives 2x² + 14x + 12; D has no 2x² term at all.
Q5. A — ⁴√(x³). denominator is the root, numerator is the power B swaps them. C treats the exponent as a division. D adds the exponents instead of using one fraction.
Educerie · written from the published College Board* Assessment Framework for the Digital SAT Suite *(Math, Advanced Math, skill/knowledge testing point "Equivalent expressions"). All questions and explanations are original Educerie text. Last reviewed 12 September 2026.
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