Probability & Counting
Basic Probability
P(event) = favorable / total
Independent Events
P(A and B) = P(A) · P(B)
Ex: Coin H & die 6 → ½·⅙ = 1/12
Permutations & Combinations
nPr = n! / (n−r)! (order matters)
nCr = n! / [r!(n−r)!] (order doesn't)
n = total items; r = items chosen
Comb: Choose 3 from 10 → 10C3 = 120
Comb: Pick 2 toppings from 5 → 5C2 = 10
Perm: Arrange 3 from 5 → 5P3 = 60
Perm: 1st/2nd/3rd from 8 → 8P3 = 336
Complex Numbers
Imaginary Unit & Powers of i
i = √(−1)
i² = −1
i¹=i, i²=−1, i³=−i, i⁴=1 (cycle of 4)
Ex: i¹⁷ → 17 mod 4 = 1 → i
Add / Multiply
(a+bi)+(c+di) = (a+c)+(b+d)i
(a+bi)(c+di) = (ac−bd)+(ad+bc)i
Dividing (Conjugate)
Multiply top & bottom by (a−bi)
Ex: (3+2i)/(1−i) × (1+i)/(1+i) = (1+5i)/2 = ½+(5/2)i
Polynomials & Factoring
Special Products
(a+b)² = a² + 2ab + b²
(a−b)² = a² − 2ab + b²
(a+b)(a−b) = a² − b²
Factoring Strategy
1. GCF first
2. Difference of squares? a²−b²=(a+b)(a−b)
3. Trinomial: find two numbers multiplying to ac & adding to b
4. Grouping for 4 terms
Remainder Theorem
f(x) ÷ (x−c) → remainder = f(c)
Ex: f(x)=x³−2x+1 ÷ (x−2) → f(2)=8−4+1=5
Sequences & Series
Arithmetic Sequence
aₙ = a₁ + (n−1)d
Sum = n/2 · (a₁ + aₙ)
a₁=first term, d=common difference (aₙ−aₙ₋₁), n=position
Ex: 3,7,11,15… d=4 → a₁₀ = 3+9(4) = 39
Geometric Sequence
aₙ = a₁ · rⁿ⁻¹
r=common ratio (aₙ/aₙ₋₁)
Ex: 2,6,18,54… r=3 → a₅ = 2·3⁴ = 162
Test-Day Strategies
Plugging In
When variables are in answer choices, pick a simple number (x=2). Evaluate, then match. Avoid 0 and 1.
Back-Solving
Start with answer choice C (middle). If too big, try smaller; if too small, try larger.
Desmos
SAT provides Desmos on Module 2. Graph both sides to find intersections. Type inequalities to visualize.
Unit Analysis
Always check units. For "what does slope mean?" → look at y-units per x-unit.
Read Twice
Traps: "find 2x not x", "value that is NOT a solution", "equivalent expression".
Time Mgmt
~1.5 min/question. Flag hard ones and move on. Easy points first.
Translating Words into Math
| Operation | Key Word / Phrase | Example | Translation | | Operation | Key Word / Phrase | Example | Translation |
|---|
| Addition (+) | plus | A number plus three | x + 3 | | Multiplication (×) | times | Eight times a number | 8x |
| more than | Ten more than a number | x + 10 | | the product of | The product of 14 and a number | 14x |
| the sum of | The sum of a number and five | x + 5 | | twice; double | Twice a number | 2x |
| increased by | A number increased by two | x + 2 | | multiplied by | A number multiplied by −6 | −6x |
| the total of | The total of six and a number | 6 + x | | of | Three fourths of a number | (3/4)x |
| added to | Eleven added to a number | x + 11 | | Division (÷) | the quotient of | The quotient of a number and 7 | x / 7 |
| Subtraction (−) | minus | A number minus seven | x − 7 | | divided by | Ten divided by a number | 10 / x |
| less than | Four less than a number | x − 4 | | the ratio of | The ratio of a number to 15 | x / 15 |
| the difference of | The difference of a number and 3 | x − 3 | | Powers (xⁿ) | the square of; squared | A number squared | x² |
| decreased by | A number decreased by twelve | x − 12 | | the cube of; cubed | A number cubed | x³ |
| subtracted from | Six subtracted from a number | x − 6 | | Equals (=) | equals; is | Seven less than a number equals ten | x − 7 = 10 |
| | is the same as | Eight is the same as twice a number | 8 = 2x |
| | amounts to / yields | x+12 yields five | x + 12 = 5 |