Algebra

Algebra is the part of mathematics that uses letters in place of numbers. That sentence sounds glib, but it captures what makes algebra both useful and, at first, difficult. The moment you write $x$ where you would have written $5$, you stop talking about a single answer and start talking about an entire pattern: every value the unknown could possibly take. From that move comes almost everything past arithmetic — school maths, physics, economics, machine learning — because the world we want to describe is rarely about one number; it is almost always about a relationship between numbers.

This section of the site covers the core algebraic techniques you need at GCSE and A-Level, on the SAT, and in foundation-level university courses: solving equations of various shapes, manipulating polynomials, and working with systems of linear equations. Each piece is one long article with a calculator embedded as a way to check your hand-work.

Articles in this section

• Quadratic equations: from intuition to the formula

  Three solving methods (factoring, completing the square, the formula), what the discriminant predicts before you ever solve, and the five mistakes I see most often when students apply the formula on a test.

  By Daniel Whitman · 16 min read

• Systems of linear equations

  Substitution, elimination, and when each one is faster on a 2×2 system. With a step-by-step solver and a worked example showing how “no solution” and “infinitely many solutions” look algebraically.

  By Daniel Whitman · 12 min read

• Polynomial factoring

  The small handful of patterns — common factor, difference of squares, sum/product, grouping — that cover almost every factoring problem you will see at school. With a calculator that factors a polynomial and tells you which pattern it used.

  By Daniel Whitman · 11 min read

A reasonable order to read them in

If algebra is new to you, start with quadratic equations. It is the topic that makes the whole subject feel concrete, and the techniques used there (factoring, completing the square, applying a general formula) reappear constantly. Polynomial factoring is the natural follow-up, since it cleans up most of the mechanical work in solving equations. Systems of equations is somewhat separate; you can come to it from either direction without missing prerequisites.

What “algebra” covers and what it does not

At school the word usually refers to elementary algebra: equations, inequalities, polynomials, and a handful of standard functions. That is what this section is about. The same word at university can mean something much broader (linear algebra, abstract algebra, ring theory) which is genuinely a different subject and is not covered here. If you are preparing for a course in linear algebra, the most useful piece of preparation from this section is the systems-of-equations article — matrix work begins exactly there.