LC Ordinary Maths cheat sheet

Quadratic formula: the roots of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac)) / 2a, valid whenever a ≠ 0.

Discriminant rule: b² − 4ac > 0 gives two real roots, = 0 gives one repeated real root, < 0 gives no real roots (two complex roots).

Root relations: for ax² + bx + c = 0 with roots α and β, α + β = −b/a and αβ = c/a.

Factor theorem: (x − k) is a factor of f(x) exactly when f(k) = 0; the remainder on dividing by (x − k) is f(k).

Simultaneous equations: solve by substitution or by elimination; a linear and a quadratic equation generally give two solution pairs.

Inequalities: multiplying or dividing by a negative number reverses the inequality sign; quadratic inequalities are solved by finding roots then testing intervals.

Laws of indices: a^m · a^n = a^(m+n), a^m / a^n = a^(m−n), (a^m)^n = a^(mn), a^0 = 1, a^(−n) = 1/a^n.

Laws of logs: log(ab) = log a + log b, log(a/b) = log a − log b, log(a^n) = n log a, and log_b a = log_c a / log_c b.

Differentiation from first principles: f′(x) = lim(h→0) [f(x + h) − f(x)] / h.

Power rule: d/dx (ax^n) = anx^(n−1); the derivative of a constant is 0 and the derivative is taken term by term.

Tangent line at x = a has slope f′(a); its equation is y − f(a) = f′(a)(x − a).

Stationary points occur where f′(x) = 0; use the sign of f″(x) (negative = local maximum, positive = local minimum) or a sign table to classify.

Product rule: (uv)′ = u′v + uv′; quotient rule: (u/v)′ = (u′v − uv′) / v²; chain rule: d/dx f(g(x)) = f′(g(x))·g′(x).

Integration reverses differentiation: ∫ x^n dx = x^(n+1)/(n+1) + C for n ≠ −1; ∫ (1/x) dx = ln|x| + C.

Definite integral: ∫ from a to b of f(x) dx = F(b) − F(a), where F is any antiderivative of f; it gives signed area under the curve.

Area between curves: integrate (upper curve − lower curve) between the intersection x-values; find intersections by setting the functions equal.

Distance between points: d = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Slope: m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes, perpendicular lines have slopes whose product is −1.

Line equations: y = mx + c (slope–intercept) and y − y₁ = m(x − x₁) (point–slope).

Circle with centre (h, k) and radius r: (x − h)² + (y − k)² = r²; expanded form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f).

Line–circle intersection: substitute the line into the circle equation; the resulting quadratic has two, one, or no real roots for secant, tangent, or no meeting.

Number sets: ℕ natural numbers, ℤ integers, ℚ rationals (p/q with q ≠ 0), ℝ real numbers; √ of a non-square prime is irrational.

Rationalising denominators: multiply by the conjugate, e.g. 1/(√a + √b) = (√a − √b)/(a − b).

Percentage and ratio work: percentage change = (change/original) × 100; a ratio a : b splits a total T into aT/(a+b) and bT/(a+b).

Probability rule: P(A) = favourable outcomes / total equally likely outcomes, with 0 ≤ P(A) ≤ 1 and P(not A) = 1 − P(A).

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events P(A ∩ B) = 0.

Multiplication rule: P(A ∩ B) = P(A)·P(B|A); for independent events P(A ∩ B) = P(A)·P(B).

Counting: number of arrangements of n distinct objects is n!; number of ways to choose r from n is nCr = n! / (r!(n − r)!).

Mean of a frequency table: x̄ = Σfx / Σf; median is the middle value (average of two middle values for even n); mode is the most frequent value.

Standard deviation: σ = √(Σ(x − x̄)²/n) for a population; a large σ means data are spread widely about the mean.

Empirical rule for roughly normal data: about 68% lie within one standard deviation of the mean and about 95% within two.

Correlation coefficient r lies between −1 and 1; r near 1 indicates strong positive linear correlation, r near 0 indicates little linear correlation.

Right-triangle ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent (SOH-CAH-TOA).

Pythagoras: in a right triangle a² + b² = c², where c is the hypotenuse; converse holds, so it tests for a right angle.

Sine rule: a/sin A = b/sin B = c/sin C; use when you know a side and its opposite angle plus one more part.

Cosine rule: a² = b² + c² − 2bc cos A; use for SAS or SSS, and rearrange to cos A = (b² + c² − a²)/(2bc).

Area of a triangle: Area = ½ab sin C, using two sides and the included angle.

Radians: π radians = 180°; arc length s = rθ and sector area A = ½r²θ when θ is in radians.

Trigonometric identity: sin²θ + cos²θ = 1, and tan θ = sin θ / cos θ.

Sine and cosine graphs: sin θ has period 360° and range [−1, 1]; cos θ also has period 360° and is the sine graph shifted 90°.

Arithmetic sequence: Tₙ = a + (n − 1)d; sum of n terms Sₙ = n/2 [2a + (n − 1)d] = n/2 (first + last).

Geometric sequence: Tₙ = ar^(n−1); sum of n terms Sₙ = a(1 − r^n)/(1 − r) for r ≠ 1.

Sum to infinity of a geometric series exists only when |r| < 1, and then S∞ = a/(1 − r).

Compound interest: A = P(1 + i)^n, where P is principal, i the rate per period as a decimal, and n the number of periods.

Reducing-balance loans use the same formula; the outstanding balance after n payments is found by accumulating the principal and subtracting the accumulated value of the payments.

Present value: the amount A due in n periods at rate i per period is worth A/(1 + i)^n today.

Recurring decimals to fractions: for 0.ababab…, let x equal the decimal, multiply by 10^(period length), subtract, and solve for x.

Angles in a triangle sum to 180°; the exterior angle of a triangle equals the sum of the two interior opposite angles.

Isosceles triangle theorem: if two sides are equal, the angles opposite them are equal, and conversely.

Parallelogram properties: opposite sides and opposite angles are equal, and diagonals bisect each other.

Cyclic quadrilateral: opposite angles sum to 180°; the angle in a semicircle is 90°.

Tangent–radius: a tangent is perpendicular to the radius at the point of contact; tangents from an external point are equal in length.

Congruence tests: SSS, SAS, ASA (with corresponding sides/angles), and RHS for right triangles; use them to prove results.

Construction steps: a perpendicular bisector is drawn with equal arcs from both endpoints; an angle bisector with equal arcs from both arms; a 60° angle from an equilateral triangle construction.

Similar triangles: corresponding angles are equal and corresponding sides are proportional; areas are in the ratio of the squares of corresponding sides.