Guides that explain the method behind common problems, with worked examples you can check in the calculators.
Algebra
A linear equation is solved by undoing what has been done to the unknown, in reverse order, while keeping both sides equal. This guide turns that idea into a five-step routine and works through every common variation.
Algebra
Factoring is fastest when it works, completing the square explains where the formula comes from, and the quadratic formula always works. Here is how to choose, with every example checked.
Algebra
Factoring is multiplication in reverse. Work through a fixed checklist, starting with the greatest common factor, and almost every polynomial you meet in algebra can be broken into simple factors.
Algebra
A system of equations asks for values that satisfy every equation at once. Substitution and elimination both work for any linear system; the art is choosing the one that needs less arithmetic.
Algebra
Simplifying means rewriting an expression in an equivalent, shorter form without changing its value for any allowed input. A handful of rules, applied in a sensible order, covers almost every case.
Algebra
Inequalities are solved almost exactly like equations, with one extra rule and a different kind of answer: a range of values instead of a single number. This guide covers every common type.
Algebra
Completing the square rewrites $x^2 + bx$ as a perfect square minus a constant. It solves any quadratic, gives the vertex of a parabola, produces the quadratic formula, and reappears in circles and calculus.
Precalculus
A logarithm answers one question: what power do I raise the base to in order to get this number? Every rule and technique in this guide follows from that single idea.
Precalculus
The domain is every input a function accepts; the range is every output it can produce. Finding the domain means hunting for forbidden inputs, and finding the range means asking which outputs are actually reachable.
Precalculus
Composing functions means feeding the output of one function into another. It is how complicated formulas are built from simple pieces, and recognising those pieces is essential for inverses and for the chain rule in calculus.
Precalculus
An inverse function undoes what the original function does. If $f$ sends 2 to 7, its inverse sends 7 back to 2. This guide explains when an inverse exists, how to find its formula, and how to check it.
Precalculus
Polynomials are the simplest functions beyond lines and parabolas, and their graphs follow a few dependable rules. Once you know the degree, the leading coefficient and the zeros, you can sketch almost any polynomial by hand.
Trigonometry
The unit circle turns trigonometry from a list of facts to memorise into a picture you can read. Every angle corresponds to a point on the circle, and that point's coordinates are the cosine and sine of the angle.
Trigonometry
A trigonometric identity is an equation that holds for every angle where both sides are defined. A short list of identities, all built from the unit circle, lets you simplify expressions, find exact values, prove results and solve equations.
Trigonometry
The law of sines says that in any triangle, each side divided by the sine of its opposite angle gives the same number. It solves triangles when you know two angles and a side, and, with care, when you know two sides and an angle opposite one of them.
Trigonometry
The law of cosines extends the Pythagorean theorem to every triangle. It finds the third side when you know two sides and the angle between them, and it finds any angle when you know all three sides.
Trigonometry
To solve a triangle is to find all three sides and all three angles from the parts you are given. Which tool to use depends on which three parts you know, so the first step is always to identify the case.
Calculus
A limit describes the value a function approaches as its input approaches some number, whether or not the function actually reaches that value. Limits are the foundation of derivatives, integrals and infinite series, so evaluating them confidently is the first real skill of calculus.
Calculus
The derivative of a function measures how fast its output changes as its input changes, at a single instant. Graphically it is the slope of the tangent line; algebraically it is the limit of average rates of change over smaller and smaller intervals.
Calculus
Computing derivatives from the limit definition is slow. A small set of rules, each proved once from that definition, lets you differentiate almost any formula in a few lines. This guide explains each rule, why it works and how to combine them.
Calculus
An integral adds up infinitely many tiny pieces to find a total: an area, a distance, a volume or any accumulated quantity. The Fundamental Theorem of Calculus links this adding-up process to antiderivatives, which is what makes integrals practical to compute.
Calculus
An indefinite integral is a family of functions, the antiderivatives, written with a constant $+ C$. A definite integral has limits of integration and produces a single number, a signed area or accumulated total. The Fundamental Theorem of Calculus lets you compute the second from the first.
Calculus
Implicit differentiation finds dy/dx for curves defined by an equation in x and y, without first solving for y. You differentiate both sides with respect to x, treat y as a function of x so the chain rule applies, and then solve for dy/dx.
Calculus
A Taylor series rewrites a function as an infinite polynomial built from its derivatives at a single point. Cutting the series off after a few terms gives a Taylor polynomial, an approximation that is easy to compute and whose error can be bounded.
Calculus
An infinite series converges when its partial sums approach a finite number. Rather than computing that number, convergence tests decide whether it exists, each test suited to a particular shape of term. Knowing which test fits which series is the real skill.
Linear Algebra
A matrix is a rectangular grid of numbers arranged in rows and columns. Matrices store data compactly and, more importantly, represent linear transformations and systems of equations. This guide covers the notation and the core operations, with special attention to matrix multiplication.
Linear Algebra
The determinant is a single number calculated from a square matrix. It tells you whether the matrix is invertible, how the matrix scales areas and volumes, and whether it flips orientation. This guide shows every standard method for computing it, with worked examples.
Linear Algebra
The inverse of a square matrix undoes what the matrix does, just as dividing undoes multiplying. A matrix has an inverse exactly when its determinant is nonzero. This guide covers the quick 2 by 2 formula, the general Gauss-Jordan method and how inverses solve systems of equations.
Linear Algebra
An eigenvector of a matrix is a direction that the matrix only stretches or flips, never turns, and the eigenvalue is the stretch factor. They reveal the essential behaviour of a matrix and underpin diagonalization, stability analysis, Markov chains and data compression.
Linear Algebra
The dot product combines two vectors into a single number that measures how much they point in the same direction. It is the tool for finding angles between vectors, checking perpendicularity, projecting one vector onto another and computing work in physics.
Linear Algebra
The cross product multiplies two vectors in three dimensions and returns a new vector perpendicular to both. Its length equals the area of the parallelogram the two vectors span, which makes it the standard tool for normal vectors, areas, torque and volumes.
Statistics
Mean, median and mode describe the centre of a data set, while range and standard deviation describe how spread out it is. Knowing how to compute each one, and when each gives an honest summary, is the foundation of statistics.
Statistics
Standard deviation measures how far data values typically lie from their mean. This guide walks through the calculation step by step, explains when to divide by n and when by n minus 1, and shows how to read the answer once you have it.
Statistics
Probability measures how likely an event is on a scale from 0 to 1. A handful of rules, for complements, unions, independent and dependent events and conditional probability, solve most problems, and knowing the classic traps prevents most mistakes.
Statistics
A z-score tells you how many standard deviations a value lies above or below the mean. It lets you compare values measured on different scales and, for normally distributed data, convert any value into a percentile or probability.
Statistics
Permutations count arrangements where order matters, and combinations count selections where it does not. Both grow from the fundamental counting principle and factorials, and they power probability calculations from card games to lotteries.
Study Guides
Math is learned by doing, not by reading. Short, regular sessions of active problem solving, spaced over time and mixed across topics, combined with understanding why each method works, build skills that last through exams and beyond.
Study Guides
Careless mistakes are rarely random. Most fall into a handful of predictable categories, such as dropped negative signs, distribution errors and misread questions, and each category has a specific habit that prevents it.
Study Guides
Almost every math answer can be checked without the answer key. Substituting back, working in reverse, estimating, testing special values and checking units catch the large majority of errors in a fraction of the time the original work took.
Study Guides
Effective exam preparation starts weeks ahead. Map what will be tested, diagnose weak topics, practise with past papers under timed conditions, plan your pacing and use the last days for review rather than new learning.
Study Guides
Not all practice is equal. Effective math practice targets specific weaknesses, mixes problem types, spaces sessions over time, varies how problems are presented and gets quick feedback, so each hour produces lasting skill rather than short-term familiarity.
Study Guides
The digital SAT Math section has 44 questions in two adaptive 35-minute modules, with a calculator allowed throughout. It tests algebra, advanced math, problem-solving and data analysis, and geometry and trigonometry, and rewards fluent algebra, smart calculator use and steady pacing.
Study Guides
The enhanced ACT Math section has 45 questions in 50 minutes, each with four answer choices, and a permitted calculator may be used throughout. It covers pre-algebra through trigonometry, so broad fluency and steady pacing matter more than any single advanced topic.
Study Guides
The AP Calculus exams combine multiple-choice and free-response sections, each worth half the score, with calculator and non-calculator parts. Success depends on fluent differentiation and integration, clear justifications and practice with released free-response questions.
Every article on this blog teaches one idea from start to finish. It begins with the definition or rule, shows the key formula, and then works through several examples in full, one line at a time, so you can see why each step is allowed rather than just what the answer is. Each guide ends with answers to the questions students ask most often about that topic.
The articles are grouped into seven categories that follow the usual order of a school and early university syllabus: algebra, precalculus, trigonometry, calculus, linear algebra, statistics and study guides. If you are reviewing for a course, start with the category that matches your current unit and read the guides in order; later articles assume the ideas from earlier ones and link back to them.
Every worked example can be checked in the matching calculator. Try each problem on paper first, then compare your steps with the solver. The worked examples in these guides are checked with automated math tests before publication, and each article shows the date it was last updated.