Mathematics Advanced

A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. Three standard solution methods:

• Factoring: rewrite as (x − r₁)(x − r₂) = 0 and set each factor to zero. Works neatly when integer roots exist.
• Completing the square: add and subtract (b/2a)² to write the left side as a perfect square.
• Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a). Always works.

The discriminant Δ = b² − 4ac determines the nature of roots:

• Δ > 0 → two distinct real roots
• Δ = 0 → one repeated real root x = −b/(2a)
• Δ < 0 → no real roots (two complex conjugate roots)

Example: solve 2x² − 7x + 3 = 0
  Δ = (−7)² − 4(2)(3) = 49 − 24 = 25
  x = (7 ± √25) / 4 = (7 ± 5) / 4
  x₁ = 12/4 = 3,   x₂ = 2/4 = 1/2

A system of linear equations is a set of two or more equations in the same variables. For two equations in two unknowns (x, y) there are three possibilities: exactly one solution (lines intersect), infinitely many solutions (same line), or no solution (parallel lines).

Solution methods:

• Substitution: solve one equation for one variable, substitute into the other.
• Elimination (addition): multiply equations so one variable cancels when added.
• Graphing: the solution is the intersection point.

Example: solve  x + 2y = 8  and  3x − y = 3
  From eq 1: x = 8 − 2y
  Substitute: 3(8 − 2y) − y = 3
    24 − 6y − y = 3  →  7y = 21  →  y = 3
  x = 8 − 2(3) = 2
  Solution: (x, y) = (2, 3)

A function f : A → B assigns to each element of the domain A exactly one element of the codomain B. The range is the set of values actually attained: {f(x) : x ∈ A}.

Key vocabulary: f is one-to-one (injective) if distinct inputs produce distinct outputs; it is onto (surjective) if every element of B is hit.

Standard graph transformations of y = f(x):

• y = f(x) + k → vertical shift up by k
• y = f(x − h) → horizontal shift right by h
• y = −f(x) → reflection over the x-axis
• y = f(−x) → reflection over the y-axis
• y = a·f(x) → vertical stretch/compression by factor |a|

Example: g(x) = √(x + 3) − 2 is f(x) = √x
  shifted left 3 units and down 2 units.
  Domain of g: x + 3 ≥ 0  →  x ≥ −3, i.e. [−3, ∞)
  Range: y ≥ −2, i.e. [−2, ∞)

A polynomial of degree n is p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ with aₙ ≠ 0. The Fundamental Theorem of Algebra guarantees exactly n roots (counted with multiplicity) in ℂ.

The Factor Theorem: (x − r) is a factor of p(x) if and only if p(r) = 0. This links roots and factors directly.

The Remainder Theorem: dividing p(x) by (x − r) leaves remainder p(r).

Polynomial long division and synthetic division allow factoring higher-degree polynomials once one root is known.

Example: p(x) = x³ − 6x² + 11x − 6
  Test x = 1: 1 − 6 + 11 − 6 = 0  ✓  → (x − 1) is a factor.
  Divide: p(x) = (x − 1)(x² − 5x + 6) = (x − 1)(x − 2)(x − 3)
  Roots: x = 1, 2, 3

An exponential function has the form f(x) = aˣ with a > 0, a ≠ 1. For a > 1 it is increasing; for 0 < a < 1 it is decreasing. The natural base is e ≈ 2.718.

The logarithm logₐ(x) is the inverse: logₐ(x) = y ⟺ aʸ = x. Key laws:

• logₐ(mn) = logₐ m + logₐ n
• logₐ(m/n) = logₐ m − logₐ n
• logₐ(mⁿ) = n logₐ m
• Change of base: logₐ x = ln x / ln a

Example: solve 3^(2x−1) = 27
  27 = 3³  →  3^(2x−1) = 3³  →  2x − 1 = 3  →  x = 2

Example: solve log₂(x + 1) + log₂(x − 1) = 3
  log₂((x+1)(x−1)) = 3  →  x² − 1 = 8  →  x² = 9  →  x = 3
  (x = −3 rejected: log requires positive argument)

An arithmetic sequence has a constant difference d between consecutive terms: aₙ = a₁ + (n−1)d. The sum of the first n terms is:

Sₙ = n(a₁ + aₙ)/2 = n(2a₁ + (n−1)d)/2

A geometric sequence has a constant ratio r: aₙ = a₁ · rⁿ⁻¹. The sum of the first n terms is:

Sₙ = a₁(1 − rⁿ)/(1 − r)   (r ≠ 1)

An infinite geometric series converges when |r| < 1:

S∞ = a₁/(1 − r)

Example: arithmetic S₁₀₀ of 1 + 2 + … + 100 = 100·101/2 = 5050.
Example: geometric 1 + 1/2 + 1/4 + … → S∞ = 1/(1 − 1/2) = 2.

In a right triangle with hypotenuse h, opposite side o, adjacent side a to angle θ:

• sin θ = o/h, cos θ = a/h, tan θ = o/a = sin θ/cos θ

On the unit circle (radius 1), the point at angle θ is (cos θ, sin θ). This extends trig to all angles.

Key values: sin 0° = 0, sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1.

The fundamental Pythagorean identity: sin²θ + cos²θ = 1. From it:

• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ

Double-angle formulas:
  sin(2θ) = 2 sin θ cos θ
  cos(2θ) = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1

The Pythagorean theorem: in a right triangle with legs a, b and hypotenuse c: a² + b² = c². Converse is also true.

Common Pythagorean triples: (3, 4, 5), (5, 12, 13), (8, 15, 17).

Circle formulas (radius r):

• Circumference: C = 2πr
• Area: A = πr²
• Arc length for central angle θ (in radians): s = rθ
• Sector area: A = (1/2)r²θ

Similar triangles have equal corresponding angles and proportional corresponding sides. If △ABC ~ △DEF with scale factor k, then each side of △DEF is k times the corresponding side of △ABC, and their areas are in ratio k².

Example: a 6-8-10 right triangle.
  6² + 8² = 36 + 64 = 100 = 10² ✓

The fundamental counting principle: if event A can occur in m ways and event B in n ways, both together can occur in m × n ways.

Permutations (order matters): P(n, r) = n!/(n−r)! ways to arrange r items from n.

Combinations (order does not matter): C(n, r) = n! / (r!(n−r)!) = _nC_r.

Probability of event E: P(E) = (number of favourable outcomes) / (total equally-likely outcomes). Rules:

• P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• P(Aᶜ) = 1 − P(A)
• If A and B are independent: P(A ∩ B) = P(A)·P(B)

Example: how many 5-card hands from a 52-card deck?
  C(52, 5) = 52!/(5!·47!) = 2 598 960

The limit of f(x) as x approaches a is the value f(x) gets arbitrarily close to, written lim_x→a f(x) = L. The function need not be defined at x = a itself.

Key limit laws (assuming lim f = L and lim g = M):

• lim [f(x) + g(x)] = L + M
• lim [f(x) · g(x)] = L · M
• lim [f(x)/g(x)] = L/M, provided M ≠ 0

A limit of 0/0 form is indeterminate. Factor and cancel or use other techniques.

Example: lim(x→2) (x² − 4)/(x − 2)
  = lim(x→2) (x+2)(x−2)/(x−2)
  = lim(x→2) (x+2)
  = 4

One-sided limits: lim(x→0⁺) 1/x = +∞,  lim(x→0⁻) 1/x = −∞
  → the two-sided limit at 0 does not exist.