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Math Concept Guides

Each guide derives the method rather than asserting it, then gives you worked examples and practice to make it stick.

Quadratic formulaFour ways to solve a quadratic, how the discriminant tells you which to reach for, and a derivation of the formula by completing the square.Factoring quadraticsRecognise GCF, difference of squares, perfect square trinomials, and grouping on sight, then apply each with worked examples and practice.Systems of equationsSubstitution, elimination, and graphing worked on the same system, with a clear rule for choosing the fastest method for the one in front of you.Chain ruleHow to spot a composite function, apply the chain rule without losing track of the inner derivative, and check the result before moving on.Product & quotient rulesWhen each rule applies, why the quotient rule has the order it does, and how to avoid the sign error that costs marks in almost every exam.Integration by partsChoose u and dv correctly with LIATE, handle repeated applications, and recognise the circular case where the integral reappears on both sides.U-substitutionHow to spot when substitution will work, choose u deliberately rather than by trial, and change the limits properly on a definite integral.Limits & continuityWhat a limit actually means before any epsilon appears, how to resolve indeterminate forms, and the three conditions for a function to be continuous.Unit circleDerive the exact values instead of memorising sixteen of them, and read any standard angle's sine and cosine straight off the circle.Trig identitiesWhich identities to know cold, which to derive on the spot, and how to choose a starting side when you are asked to prove one from scratch.Word problemsA five-step framework for turning a word problem into algebra, including the phrase-to-operator table and the 'less than' reversal trap.Order of operationsWhy PEMDAS misleads people on division and subtraction, and how to read an expression the way a marker does when awarding method marks.Exponent rulesEvery exponent rule derived from a single idea, with negative and fractional exponents shown to follow the same pattern rather than new rules.Standard deviationCompute it by hand step by step, understand what it actually measures about spread, and know exactly when to divide by n rather than n minus 1.Matrix inverseThe adjugate method and Gauss-Jordan elimination compared on the same matrix, with a clear test for when an inverse does not exist at all.

Where to start

These guides are ordered by dependency rather than difficulty. If calculus is not landing, the problem is usually not calculus — it is almost always factoring, exponent rules, or function notation quietly failing underneath. Start there and the derivative rules stop feeling arbitrary.

Each guide derives its result rather than asserting it. That is deliberate: a formula you have watched being built is one you can reconstruct in an exam when memory fails, and a formula you have only memorised is one you will misapply the moment the problem is phrased unusually. The unit circle guide is the clearest example — there are sixteen values in the table and only two you actually need to remember.

Every guide ends with practice you can check in place, and links to the solver for the subject so you can put your own problem through the same method.

Put it into practice

Subject solversGet a worked solution to your own problem.All calculatorsCompute with the steps shown.Reading the blogLonger pieces on studying and method.How to check your workSix ways to catch your own mistakes.

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