Algebra 2 — Essential Formulas

Algebra 2 CBE essentials — quadratics, exponentials, logs, complex numbers, sequences, and function transformations on one page.

Quadratics

• Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a)
• Discriminant: b² − 4ac → positive (2 real), zero (1 real), negative (2 complex)
• Vertex form: y = a(x − h)² + k → vertex at (h, k)
• Axis of symmetry: x = −b/(2a)
• Sum/product of roots: −b/a and c/a

Complex Numbers

• i² = −1; i³ = −i; i⁴ = 1
• (a + bi)(c + di) = (ac − bd) + (ad + bc)i
• Conjugate: (a + bi)(a − bi) = a² + b²

Exponents & Logarithms

• log_b(xy) = log_bx + log_by
• log_b(x/y) = log_bx − log_by
• log_b(xⁿ) = n · log_bx
• Change of base: log_bx = log x / log b
• Exponential growth: y = a·b^t (b > 1); decay: 0 < b < 1
• Continuous: y = a · e^rt

Polynomial Division

• Remainder theorem: P(c) = remainder when P(x) ÷ (x − c)
• Factor theorem: P(c) = 0 ⇔ (x − c) is a factor
• Rational root theorem: any rational root = ±(factor of constant) / (factor of leading coeff)

Rational Expressions

• Find a common denominator before adding/subtracting
• Watch for extraneous solutions when squaring or cross-multiplying
• Vertical asymptote: where denominator = 0 (and numerator ≠ 0)

Function Transformations

• y = f(x) + k: shift up k
• y = f(x − h): shift right h
• y = −f(x): reflect across x-axis
• y = f(−x): reflect across y-axis
• y = a·f(x): vertical stretch (|a| > 1) or compress

Sequences & Series

• Arithmetic: a_n = a₁ + (n − 1)d; sum = n/2 · (a₁ + a_n)
• Geometric: a_n = a₁ · rⁿ⁻¹; sum = a₁(1 − rⁿ) / (1 − r)
• Infinite geometric (|r| < 1): S = a₁ / (1 − r)

Common Test Mistakes

• Forgetting ± when taking a square root
• Dropping extraneous roots after squaring
• Mixing up log rules with exponent rules
• Using arithmetic formula on a geometric sequence (or vice versa)

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