SAT Math · Advanced Math · Skill 1 of 3

Equivalent expressions

Expand, factor and rewrite expressions so they look different but stay equal for every value of x. This is the toolkit behind almost every Advanced Math question.

9 chapters plus a challengeAbout 40 minutesFree, no sign-up

Start with the area model →Jump to the challenge

Hi, I’m Io. Same value, different outfit. That is all an equivalent expression is.

Tap Io

See expansion as area

Explore · 4 min

A rectangle with sides x + a and x + b splits into four tiles: x², ax, bx and ab. Add the tiles and you have the expanded form. The two x-tiles combine, which is why the middle number is a + b and the last number is a × b.

Expand and combine like terms

Practise · 6 min

To expand, multiply every term by every term, then combine like terms (terms with the same power of x). Watch the signs: a negative outside a bracket changes the sign of everything inside.

Worked example: 3(2x − 5) − 2(x − 4)

Distribute the 3: 6x − 15
Distribute the −2 to both terms: −2x + 8 (−2 × −4 = +8)
Combine like terms: 6x − 2x = 4x and −15 + 8 = −7, so the answer is 4x − 7

Factoring: expanding in reverse

Game · 6 min

Factoring rewrites a sum as a product. Check these patterns in order:

Common factor: 6x² + 9x = 3x(2x + 3)
Difference of squares: x² − 49 = (x − 7)(x + 7)
Perfect square: x² + 10x + 25 = (x + 5)²
Other trinomials: x² − x − 12 = (x − 4)(x + 3). Find two numbers that multiply to −12 and add to −1.

Sort by pattern

Three special products

Learn · 4 min

These three show up constantly. Knowing them by sight saves time in both directions, expanding and factoring.

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

(a + b)(a − b) = a² − b²

Watch out(x + 4)² is not x² + 16. Write it as (x + 4)(x + 4) and you will see the 8x appear.

Exponent rules and radicals

Practise · 5 min

Exponent questions on the SAT test a short list of rules. A radical is just a fractional exponent in disguise: the root goes in the denominator.

x^a · x^b = x^{a + b}
(x^a)^b = x^ab
x^a ÷ x^b = x^{a − b}
x^−a = 1 ÷ x^a
√x = x^1/2, and the cube root of x² is x^2/3

Rational expressions

Learn · 5 min

A rational expression is a fraction with polynomials. Three moves cover most SAT questions: factor and cancel, use a common denominator to add, and rewrite a division as a whole part plus a remainder.

Adding: 2/x + 3/(x + 1)

The common denominator is x(x + 1)
Rewrite each fraction: 2(x + 1)/(x(x + 1)) + 3x/(x(x + 1))
Add the tops: 2x + 2 + 3x = 5x + 2, so the sum is (5x + 2)/(x(x + 1))

Dividing: (x² + 5x + 7) ÷ (x + 2)

Find the closest product you know: (x + 2)(x + 3) = x² + 5x + 6
The top is 1 more than that: x² + 5x + 7 = (x + 2)(x + 3) + 1
Divide each part by (x + 2): x + 3 + 1/(x + 2)

Match the coefficients

Explore · 5 min

A favourite SAT format: “if this equals that for all values of x, find the constant.” Expand one side, then set the x² numbers equal, the x numbers equal and the constants equal. Use the sliders to find a and b.

Check it in Desmos

Tools · 3 min

The Digital SAT has Desmos built in. Equivalent expressions give the same graph, so graphing both sides is a fast check when you are unsure.

Spot the trap

Detective · 3 min

Each solution has one faulty step. These are the slips that SAT wrong answer choices are built from.

Final challenge

8 questions · 10 min

Key ideas to carry in

Every term by every termThen combine like terms.

Factor in orderCommon factor, difference of squares, then trinomials.

Special products(a + b)² has a middle term 2ab.

ExponentsMultiply: add. Power of a power: multiply. Root: fraction.

Cancel factors onlyFactor first, then cancel.

For all xMatch the coefficients of each power.

Challenge