Algebra 2 Master Workbook

Unit 01 · Quadratic Functions

Standard, Vertex & Factored Forms

Standard: f(x) = ax² + bx + c

Vertex Form: f(x) = a(x − h)² + k → Vertex: (h, k)

Factored: f(x) = a(x − r₁)(x − r₂)

🔑 Quadratic Formula: x = [−b ± √(b²−4ac)] / 2a

🔑 Discriminant (b²−4ac): >0 → 2 real roots | =0 → 1 real | <0 → 2 complex

Example

Solve x² − 5x + 6 = 0
Factor: (x−2)(x−3) = 0 → x = 2 or x = 3

Unit 02 · Polynomial Operations

Remainder & Factor Theorems

🔑 Remainder Theorem: If p(x) is divided by (x−c), remainder = p(c)

🔑 Factor Theorem: (x−c) is a factor of p(x) if and only if p(c) = 0

End behavior: determined by leading term axⁿ

Example

Find remainder when p(x) = x³ − 2x + 1 is divided by (x−2)
p(2) = 8 − 4 + 1 = 5

Unit 03 · Rational Expressions

Simplifying, Operations & Asymptotes

🔑 Vertical Asymptote: Set denominator = 0 (after simplifying)

🔑 Horizontal Asymptote: Compare degrees of numerator (n) & denominator (d):
n < d → y = 0 | n = d → y = leading coeff. ratio | n > d → none (oblique)

Example

f(x) = (x²−1)/(x−1) = (x+1)(x−1)/(x−1) = x+1, x≠1
Hole at x = 1, not a vertical asymptote

Unit 04 · Radicals & Complex Numbers

Radical Operations & Imaginary Units

i = √(−1), i² = −1, i³ = −i, i⁴ = 1 (cycle of 4)

🔑 Complex Number: a + bi where a = real, b = imaginary

🔑 Conjugate: (a+bi)(a−bi) = a²+b² (always real)

Example

(3+2i)(1−i) = 3−3i+2i−2i² = 3−i+2 = 5−i

Unit 05 · Exponentials & Logarithms

Log Properties & Change of Base

logb(MN) = logb(M) + logb(N)

logb(M/N) = logb(M) − logb(N)

logb(Mⁿ) = n·logb(M)

🔑 Change of Base: logb(x) = ln(x)/ln(b) = log(x)/log(b)

🔑 Inverse relationship: b^(logb x) = x and logb(b^x) = x

Example

Solve log₂(x+3) = 4
x+3 = 2⁴ = 16 → x = 13

Unit 06 · Sequences & Series

Arithmetic & Geometric Sequences

Arithmetic: aₙ = a₁ + (n−1)d | Sₙ = n(a₁+aₙ)/2

Geometric: aₙ = a₁·rⁿ⁻¹ | Sₙ = a₁(1−rⁿ)/(1−r)

🔑 Infinite Geo. Sum: S∞ = a₁/(1−r), only if |r| < 1

Example

Arithmetic seq: a₁=3, d=4 → a₁₀ = 3 + 9(4) = 39

Unit 07 · Conic Sections

Circle, Parabola, Ellipse, Hyperbola

Circle: (x−h)²+(y−k)² = r²

Ellipse: x²/a² + y²/b² = 1 (a>b, horizontal major axis)

Hyperbola: x²/a² − y²/b² = 1

🔑 Parabola: vertex form y = a(x−h)²+k; focus = h, k+1/(4a)

Example

x²+y²−6x+4y−3=0 → complete square:
(x−3)²+(y+2)² = 16 → center (3,−2), r=4

Unit 08 · Matrices

Operations, Determinant & Inverse

🔑 Matrix Multiplication: A(m×n)·B(n×p) = C(m×p)

det([[a,b],[c,d]]) = ad − bc

🔑 Inverse: A⁻¹ = (1/det(A))·[[d,−b],[−c,a]]

Example

A = [[2,3],[1,4]] → det = 8−3 = 5
A⁻¹ = (1/5)[[4,−3],[−1,2]]

Unit 09 · Systems of Equations

Linear & Nonlinear Systems

🔑 Methods: Substitution, Elimination, Matrices (Cramer's Rule)

🔑 Nonlinear: Substitute and solve the resulting equation — may yield 0, 1, or 2+ solutions

Example

y = x² and y = x+2
x² = x+2 → x²−x−2=0 → (x−2)(x+1)=0
x=2 or x=−1 → (2,4) and (−1,1)

Unit 10 · Binomial Theorem

Binomial Expansion & Pascal's Triangle

(a+b)ⁿ = Σ C(n,k)·aⁿ⁻ᵏ·bᵏ, k=0 to n

🔑 C(n,k) = n! / [k!(n−k)!]

🔑 (k+1)th term: C(n,k)·aⁿ⁻ᵏ·bᵏ

Example

3rd term of (x+2)⁵:
C(5,2)·x³·2² = 10·x³·4 = 40x³

Algebra 2Master Workbook

📚 Core Concepts & Key Formulas

Algebra 2
Master Workbook