11.1 Equivalent Expressions
Key Takeaways
- Two expressions are equivalent when they give the same value for every allowed input; expand or factor until the structures match, then check with a number such as x = 1 or x = 2.
- (a + b)^2 = a^2 + 2ab + b^2, so (x + 6)^2 = x^2 + 12x + 36, not x^2 + 36; the middle term is twice the product.
- Factor a GCF first (6x^2 + 9x = 3x(2x + 3)); difference of squares is a^2 - b^2 = (a - b)(a + b), so 4x^2 - 25 = (2x - 5)(2x + 5).
- Combine like terms by degree: 5x^2 - 3x + 2x^2 + 7x - 4 = 7x^2 + 4x - 4. Unlike powers never merge.
- Advanced Math is about 20% of operational PSAT 8/9 Math (7–9 questions). Rational expressions with x in a denominator are not this test; later SAT Suite tests include them.
Advanced Math is the second Math domain on the PSAT 8/9. College Board's Assessment Framework for the Digital SAT Suite puts Advanced Math at about 20% of operational Math, 7–9 questions. This independent OpenExamPrep chapter teaches all three Advanced Math skill/knowledge testing points published for PSAT 8/9: equivalent expressions, nonlinear equations in one variable (and elementary systems that include a nonlinear equation), and nonlinear functions. OpenExamPrep is not a College Board partner and does not claim official approval, review, partnership, or exact equivalence with College Board materials. Use the Student Guide, Bluebook practice, and this guide together.
Two exclusions matter on this test. Rational equations (the unknown in a denominator) and radical equations (the unknown under a root that you isolate) are not represented on PSAT 8/9. They appear on later SAT Suite tests. This chapter still uses square roots of perfect squares when a quadratic is already in a form such as x^2 = 49, and it still uses numeric fractions as coefficients. That is ordinary grade 8 algebra, not solving a rational equation.
Algebra, the larger domain at about 42.5%, stays on first-power equations and lines (9.1 Linear Equations in One Variable, 9.3 Linear Functions). Advanced Math asks you to rewrite second-power, polynomial, and compact square expressions and to work with curves. A calculator is allowed on every Math item, including Desmos. Fluency still matters: Desmos graphs whatever you type, including a dropped middle term.
What equivalent means
Two expressions are equivalent when they produce the same value for every allowed input. (x + 3)(x + 5) and x^2 + 8x + 15 are equivalent. So are 2(3x - 4) and 6x - 8. Matching a factored form to an expanded form, or a compact square to a trinomial, is the core skill.
| Move | Example | Equivalent result |
|---|---|---|
| Combine like terms | 3x^2 + 5x - 2x^2 + 4x | x^2 + 9x |
| Distribute | 4(2x - 3) | 8x - 12 |
| Expand a product | (x + 3)(x + 5) | x^2 + 8x + 15 |
| Expand a square | (x + 4)^2 | x^2 + 8x + 16 |
| Factor a GCF | 6x^2 + 9x | 3x(2x + 3) |
| Difference of squares | x^2 - 25 | (x - 5)(x + 5) |
Factored form makes zeros easy to see later. Expanded form is easier to add or to read a y-intercept of a related function. Vertex-style reading shows up in 11.3 Nonlinear Functions. Here the job is rewrite without changing the value.
Worked example: combine like terms
Simplify 3x^2 + 5x - 2x^2 + 4x - 7.
Group same powers: (3x^2 - 2x^2) + (5x + 4x) - 7.
x^2 + 9x - 7
Check with x = 2. Original: 3(4) + 5(2) - 2(4) + 4(2) - 7 = 12 + 10 - 8 + 8 - 7 = 15. Simplified: 4 + 18 - 7 = 15.
Unlike terms stay unlike. 3x^2 and 5x do not merge. Degree 2 and degree 1 are different kinds of term, the same way 3 meters and 5 seconds do not add.
Second: 7y - 4 + 2y^2 - y + 9 - 5y^2.
y^2 terms: 2y^2 - 5y^2 = -3y^2. y terms: 7y - y = 6y. Constants: -4 + 9 = 5.
-3y^2 + 6y + 5
Check y = 1: original 7 - 4 + 2 - 1 + 9 - 5 = 8. Simplified: -3 + 6 + 5 = 8.
Worked example: distribute, then combine
Simplify 4(2x - 3) + 5(x + 1).
4 · 2x = 8x, 4 · (-3) = -12, 5 · x = 5x, 5 · 1 = 5.
8x - 12 + 5x + 5 = 13x - 7
Check x = 3. Original: 4(6 - 3) + 5(3 + 1) = 4(3) + 5(4) = 12 + 20 = 32. Simplified: 13(3) - 7 = 39 - 7 = 32.
A common grade 8 error is writing 4(2x - 3) as 8x - 3. The 4 must hit both terms. If a check fails, look there first.
Second: 3(x - 2) - 2(4x + 1) = 3x - 6 - 8x - 2 = -5x - 8.
The minus in front of 2(4x + 1) distributes onto both 4x and +1. Check x = 0: original 3(-2) - 2(1) = -6 - 2 = -8. Simplified: -8. Check x = 2: original 3(0) - 2(8 + 1) = -18. Simplified: -10 - 8 = -18.
Worked example: expand (x + a)(x + b)
Expand (x + 3)(x + 5).
Distribute twice: x(x + 5) + 3(x + 5) = x^2 + 5x + 3x + 15 = x^2 + 8x + 15.
Check x = 1: (4)(6) = 24, and 1 + 8 + 15 = 24. Check x = 0: 3 · 5 = 15, matching the constant term.
Second: (x + 2)(x - 7) = x^2 - 7x + 2x - 14 = x^2 - 5x - 14.
Check x = 2: (4)(-5) = -20, and 4 - 10 - 14 = -20.
Third: (2x + 1)(x + 4) = 2x · x + 2x · 4 + 1 · x + 1 · 4 = 2x^2 + 8x + x + 4 = 2x^2 + 9x + 4.
Check x = 1: (3)(5) = 15, and 2 + 9 + 4 = 15. Check x = 2: (5)(6) = 30, and 8 + 18 + 4 = 30.
Worked example: expand (a + b)^2
The identity is (a + b)^2 = a^2 + 2ab + b^2. The middle term is twice the product. Dropping it is the most common trap: (x + 4)^2 is not x^2 + 16.
(x + 4)^2 = x^2 + 2(x)(4) + 16 = x^2 + 8x + 16.
Same work as (x + 4)(x + 4) = x^2 + 4x + 4x + 16. Check x = 3: 7^2 = 49, and 9 + 24 + 16 = 49.
(x - 5)^2 = x^2 - 10x + 25. The middle sign follows the minus: 2(x)(-5) = -10x. Check x = 5: 0^2 = 0, and 25 - 50 + 25 = 0. Check x = 6: 1^2 = 1, and 36 - 60 + 25 = 1.
(2x + 3)^2 = (2x)^2 + 2(2x)(3) + 9 = 4x^2 + 12x + 9. Check x = 1: 5^2 = 25, and 4 + 12 + 9 = 25.
(x + 6)^2 = x^2 + 12x + 36. The 12 is 2 · 6, not 6, and the 36 is 6^2, not 2 · 6. Check x = 1: 7^2 = 49, and 1 + 12 + 36 = 49.
If a stem shows 3(x + 2)^2, expand the square first, then multiply: 3(x^2 + 4x + 4) = 3x^2 + 12x + 12. Check x = 0: 3(2)^2 = 12, matching the constant. Check x = 2: 3(4)^2 = 48, and 12 + 24 + 12 = 48.
(3x - 1)^2 = 9x^2 - 6x + 1. Trap forms: 9x^2 + 1 (missing -6x), 9x^2 - 3x + 1 (used 3 instead of 2 · 3x · 1), 3x^2 - 6x + 1 (squared the 3 into the wrong place). Check x = 1: (3 - 1)^2 = 4, and 9 - 6 + 1 = 4.
Worked example: factor a GCF
Factoring undoes distributing. Pull out the greatest common factor (GCF) of every term.
6x^2 + 9x. GCF is 3x.
3x(2x + 3)
Check: 3x · 2x = 6x^2, 3x · 3 = 9x.
12x^3 - 18x^2. GCF is 6x^2.
6x^2(2x - 3)
Check: 6x^2 · 2x = 12x^3, 6x^2 · 3 = 18x^2, so 6x^2(2x - 3) = 12x^3 - 18x^2.
4x^2 + 10x - 6. GCF is 2: 2(2x^2 + 5x - 3). If the leftover quadratic factors further, that is extra; on PSAT 8/9, pulling a clean GCF is already an equivalent-form win. Check x = 1: original 4 + 10 - 6 = 8, and 2(2 + 5 - 3) = 2(4) = 8.
Worked example: difference of squares
a^2 - b^2 = (a - b)(a + b). This pattern is appropriate for grades 8 and 9 and is worth automatic recognition.
x^2 - 25 = x^2 - 5^2 = (x - 5)(x + 5).
Check: (x - 5)(x + 5) = x^2 + 5x - 5x - 25 = x^2 - 25. Check x = 6: 36 - 25 = 11, and (1)(11) = 11.
x^2 - 49 = (x - 7)(x + 7).
4x^2 - 9 = (2x)^2 - 3^2 = (2x - 3)(2x + 3).
Check: 2x · 2x = 4x^2, 2x · 3 = 6x, -3 · 2x = -6x, -3 · 3 = -9. The middle cancels.
9x^2 - 16 = (3x - 4)(3x + 4). Check x = 2: 36 - 16 = 20, and (6 - 4)(6 + 4) = 2 · 10 = 20.
4x^2 - 25 = (2x - 5)(2x + 5). Check x = 3: 36 - 25 = 11, and (6 - 5)(6 + 5) = 1 · 11 = 11.
x^2 + 25 does not factor over the reals as a difference of squares. Difference needs a minus between two perfect squares. A sum of squares is a different expression.
Worked example: factor a simple trinomial
x^2 + 8x + 15. Two numbers that multiply to 15 and add to 8: 3 and 5.
(x + 3)(x + 5)
That matches the expand example above. Check x = 0: 15 = 3 · 5.
x^2 + 7x + 12 = (x + 3)(x + 4) because 3 · 4 = 12 and 3 + 4 = 7. Check x = 1: 1 + 7 + 12 = 20, and (4)(5) = 20.
x^2 - 5x + 6 = (x - 2)(x - 3) because (-2)(-3) = 6 and -2 + -3 = -5. Check x = 4: 16 - 20 + 6 = 2, and (2)(1) = 2.
x^2 - x - 12 = (x - 4)(x + 3) because -4 · 3 = -12 and -4 + 3 = -1. Check x = 4: 16 - 4 - 12 = 0, and (0)(7) = 0.
If the leading coefficient is not 1, start with a GCF when you can: 2x^2 + 10x + 12 = 2(x^2 + 5x + 6) = 2(x + 2)(x + 3). Check x = 0: 12, and 2(2)(3) = 12.
Matching equivalent forms on a question
A typical item shows one form and asks which choice is equivalent. Expand or factor until the structure matches. Do not pick a choice that matches only one coefficient.
Which is equivalent to 2(x - 1)(x + 4)?
First (x - 1)(x + 4) = x^2 + 4x - x - 4 = x^2 + 3x - 4.
Then 2(x^2 + 3x - 4) = 2x^2 + 6x - 8.
Trap expressions: x^2 + 3x - 4 (forgot the 2), 2x^2 + 8x - 8 (used 4 as the middle without combining 4x and -x), 2x^2 + 3x - 4 (multiplied only the x^2). Check x = 2: 2(1)(6) = 12, and 8 + 12 - 8 = 12.
Polynomials, not rational expressions
A polynomial is a sum of terms a_n x^n + ... + a_1 x + a_0 with whole-number powers. Linear, quadratic, and cubic polynomials (degrees 1, 2, and 3) are in scope for rewriting. Adding (x^2 + 3x - 1) + (2x^2 - 5) = 3x^2 + 3x - 6 is equivalent-expression work. Check x = 1: original (1 + 3 - 1) + (2 - 5) = 3 - 3 = 0, and 3 + 3 - 6 = 0.
Dividing by a variable, writing 1/x, or combining 1/(x + 1) + 1/(x - 1) is rational-expression work and is not a PSAT 8/9 Advanced Math item. Keep denominators as numbers (clearing x/3 + x/6 lives in Algebra) rather than as expressions that contain x.
Desmos and SPR habits
Graph y = (left expression) and y = (right expression). If they overlap as the same curve, the rewrite is equivalent. If they are parallel or meet at one point only, you dropped a term. For (x + 4)^2 versus x^2 + 16, the graphs are different parabolas; that visual is a useful check after you already expanded on paper.
Student-produced response items may ask for a coefficient: in the equivalent form ax^2 + bx + c of (x + 6)^2, b is 12, not the whole trinomial. Type 12. Do not type x = 12. SPR entry rules live in 15.1 Student-Produced Response Questions.
Which expression is equivalent to (x + 6)^2?
Which factorization is equivalent to 4x^2 - 25?
What is 5x^2 - 3x + 2x^2 + 7x - 4 after like terms are combined?