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Math Guides with Worked Examples

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

Quadratic formulaLearn four ways to solve a quadratic and how the discriminant tells you which to use. Includes a derivation by completing the square.Factoring quadraticsSpot common factors, difference of squares, perfect square trinomials, and grouping. Each pattern is shown with worked examples.Systems of equationsSubstitution, elimination, and graphing worked on the same system, with a clear rule for choosing the fastest method.Chain ruleLearn how to spot a composite function, apply the chain rule, and keep the inner derivative. Check the result before you move on.Product & quotient rulesSee when the product rule and quotient rule apply, why the quotient rule has that order, and how to avoid the common sign error.Integration by partsChoose u and dv with LIATE, handle repeated applications, and recognise the case where the integral comes back on both sides.U-substitutionLearn when substitution works, how to choose u on purpose, and how to change the limits on a definite integral.Limits & continuityUnderstand what a limit means, how to resolve indeterminate forms, and the three conditions for a function to be continuous.Unit circleDerive the exact sine and cosine values instead of memorising a table, and read any standard angle straight off the circle.Trig identitiesLearn which trig identities to memorise, which to derive, and how to choose a starting side when you have to prove one.Word problemsA five step method for turning a word problem into algebra, including the phrase to operator table and the less than reversal.Order of operationsSee why PEMDAS misleads people on division and subtraction, and how to read an expression the way method marks are awarded.Exponent rulesEvery exponent rule from one idea. Negative and fractional exponents follow the same pattern, with radicals included.Standard deviationCalculate standard deviation by hand, see what it measures, and know when to divide by n and when to divide by n minus 1.Matrix inverseCompare the adjugate method and Gauss Jordan elimination on the same matrix, and test when an inverse does not exist.

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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