Topics

Worked examples — one figure, ~500 words, FAQs. Pick a branch.

algebra

The quadratic formula: x = (−b ± √(b² − 4ac)) / 2aWhere the formula comes from, what its three pieces mean, and when each gives you two real roots, one repeated root, or none.
Polynomial long division: (x³ − 2x² + 4x − 8) ÷ (x − 2)The systematic recipe for dividing one polynomial by another, written out the same way you divided whole numbers in primary school.
Omar Khayyam's geometric solution to the cubic x³ + bx = cSix centuries before Cardano, the 11th-century Persian poet-mathematician solved every positive-coefficient cubic by intersecting a parabola and a circle.

The chain rule: differentiating composite functionsDifferentiate f(g(x)) by multiplying f'(g(x)) and g'(x) — the most-used rule in all of calculus.
Integration by parts: ∫ u dv = uv − ∫ v duThe integration counterpart of the product rule — the formula that handles x·eˣ, x·sin(x), x²·ln(x), and a hundred other classroom integrals.
Taylor series of sin x near 0A polynomial that hugs sin(x) closer and closer as you add more terms — the bridge between calculus and approximation.

DFA construction: build an automaton for (a|b)* ending in 'ab'A minimal three-state deterministic finite automaton that accepts exactly the strings ending in 'ab' — and the systematic way to design it.
BFS vs DFS: two ways to traverse a graphSame graph, two visit orders — breadth-first sweeps level-by-level, depth-first dives down each branch first.

The Pythagorean theorem: a² + b² = c²Why the squares on the two short sides of a right triangle add up to the square on the long one — with the picture that makes it obvious.
Heron's formula: area of a triangle from its three sidesA two-thousand-year-old formula that gives the area of any triangle directly from the side lengths — no height needed.

Matrix inverse of a 3×3: the adjugate-and-determinant methodStep-by-step inverse of a 3×3 matrix using cofactors — what the formula does, why each step is necessary, and how to spot when a matrix has no inverse.
Vector projection: dropping a onto bThe component of one vector that lies along another — the geometric move behind least-squares regression, Gram-Schmidt, and almost every dot-product application.
Eigenvalues of a 2×2 matrix from det(A − λI) = 0The two scalars that survive when a matrix acts on its eigenvectors — found by solving one quadratic equation.

The 68–95–99.7 rule for the normal distributionWhy almost every observation from a normal distribution sits within three standard deviations of the mean — and how the percentages stack up.
Binomial PMF: P(X = k) = C(n, k) · pᵏ · (1−p)ⁿ⁻ᵏThe probability of getting exactly k successes in n independent trials — drawn as a bar chart for n = 10 fair-coin flips.

The unit circle: sin, cos, tan at standard anglesOne circle of radius 1 pins down every trig value you'll meet at school or in first-year university.