Engineering↗
Completing the Square
Turning any quadratic into a perfect square plus a constant, the one rule that makes it work, and why the vertex falls out of the result for free.
Content type
Guides
Structured, multi-part guides that take a subject from first principles to applied practice. 2916 pages.
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Complex Numbers: Arithmetic and the Conjugate
The imaginary unit, why it was needed, arithmetic with a + bi, the powers of i, and dividing by a complex number using its conjugate.
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Conditional and Identical Equations
The difference between an equation true for some values and one true for all of them, why only the first kind is worth solving, and what a solution set actually is.
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Consistent, Inconsistent and Dependent Systems
What it means when the variables all vanish, how to tell an impossible system from one with infinitely many solutions, and how to write a general solution.
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Corrections and Judgement Calls on the Source Notes
Every place where the teaching notes are arithmetically wrong, internally inconsistent or incomplete, what was done about it, and how each error could have been caught.
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Cubic and Higher-Degree Equations
Finding one root, dividing it out, and finishing with the quadratic that remains - plus what to do when the cubic has no rational root.
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Determinants, Minors and Cofactors
A single number that decides whether a matrix is invertible, the 2 times 2 rule, and expansion by minors and cofactors for larger matrices.
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Even and Odd Functions
Two symmetry conditions, how to test for them in one substitution, and why most functions are neither.
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Exact Values, Reference Angles and Quadrant Signs
The two special triangles, the ASTC sign rule, and the reference angle method that reduces any angle to an acute one.
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Factoring Polynomials: the Five Standard Methods
Common factors, grouping, reverse FOIL, the special forms and perfect squares - the five methods, the order to try them in, and what to do when none of them works.
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Finding the Vertex and Graphing a Quadratic
Two routes to the turning point, the four questions a complete graph answers, and a worked sketch from start to finish.
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Foundation Mathematics: the Study Pathway
How the collection fits together, which order to work through it in, and where each stream leads.
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Fractional Exponents and Radicals
Why x1/2 must mean sqrt(x), how n-th roots and fractional powers are the same thing, when a root is not a real number, and why (x6)1/2 is abs{x3} rather than x3.
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Functions: Domain, Range and Notation
The one-output rule that defines a function, the vertical line test, function notation, and what domain and range actually name.
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Gauss-Jordan Reduction
Carrying the elimination further so the solution is read directly from the matrix, with no back-substitution at all.
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Gaussian Elimination and Row-Echelon Form
Detaching the coefficients into an augmented matrix, the three legitimate row operations, and reducing to row-echelon form.
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Graphs of the Trigonometric Functions
The sine and cosine waves, the tangent curve with its asymptotes, and what the shapes reveal about domain, range and symmetry.
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Inequalities and Interval Notation
Order on the real line, the two notations for describing an interval, and the one rule that makes solving an inequality different from solving an equation.
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Literal Equations and Formula Rearrangement
Making a chosen symbol the subject of a formula, why the target must be isolated as a common factor, and the conditions that come with dividing by a variable.
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Mathematical Notation and Symbols: a Reference
Every symbol used across this collection, grouped by area, with its reading and a note on where it is easily confused.
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Matrices: Order, Equality, Addition and Scalar Multiplication
What a matrix is, how its entries are indexed, when two are equal, and the arithmetic that works entry by entry.
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Matrix Inverses and Solving Linear Systems
The matrix that undoes multiplication, the 2 times 2 formula, the Gauss-Jordan method for larger cases, and solving Ax = b directly.
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Matrix Multiplication and Matrix Algebra
Row into column, why the inner dimensions must agree, and the two familiar laws that fail.
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Multiplying Polynomials and the Special Products
Multiplying polynomials by the distributive law, the FOIL bookkeeping aid, and the seven special products worth recognising instantly in both directions.
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Parallel and Perpendicular Lines
Equal slopes for parallel, negative reciprocals for perpendicular, why the second rule holds, and the vertical-horizontal exception.
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Polynomial Functions and Their Graphs
Degree, leading coefficient and what they determine, the family of power functions y = xn, and how the special cases fit together.
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Polynomial Long Division
The division algorithm for polynomials, laid out beside the arithmetic long division it copies, with placeholders, remainders and the quotient-plus-remainder form.
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Polynomials: Terminology, Degree and Standard Form
Monomials, binomials and trinomials, how the degree of a multi-variable term is counted, what makes two terms alike, and why standard form is worth the trouble.
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Pythagoras' Theorem and the Cut-and-Paste Proof
The theorem behind every distance calculation, and a proof by rearranging four copies of a triangle inside a square.
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Quadratic Functions and the Parabola
The parabola as the graph of a second-degree function, how the leading coefficient decides its direction, and reading the range from the turning point.
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Radian Measure, Degrees, Minutes and Seconds
The radian defined by arc length, converting between the two systems, and the sexagesimal subdivision of the degree.
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Radical Equations and Spurious Solutions
Squaring both sides to remove a radical, why that can create solutions that were never there, and why checking is not optional.
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Rational Algebraic Fractions
Algebraic fractions, the values that must be excluded, reducing by cancelling factors rather than terms, and clearing compound fractions by a single multiplication.
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Rationalising Denominators and Conjugates
Clearing a radical from a denominator, the conjugate pair and why it works, and the standard forms for one-term and two-term denominators.
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Sign Diagrams for Quadratic and Rational Inequalities
Why a quadratic inequality cannot be solved by the methods that work on a linear one, and how a sign table settles it in three steps.
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Slope of a Straight Line
Slope as rise over run, why it comes out the same whichever two points are used, and what its sign and size tell you.
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Solving Linear Equations
Isolating the unknown, clearing fractions before anything else, and the transposition shortcut - with every step justified by an inverse operation.
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Solving Triangles: the Sine and Cosine Rules
Finding every side and angle of a triangle from partial information - right triangles by ratio, the rest by the sine and cosine rules.
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Symmetry Tests and Curve Sketching
Four algebraic tests that reveal a curve's symmetry before any points are plotted, and how each one halves the work of sketching.
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Systems in Three Variables
Extending elimination to three unknowns, the order in which variables are removed, and back-substitution.
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Systems of Linear Equations: Substitution and Elimination
Two equations in two unknowns, the two elementary methods, and the geometric meaning of a solution as an intersection.
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The Absolute Value Function and Its Graph
The V-shaped graph of y = |x|, how shifts and reflections move it, and a method for sketching any transformed version.
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The Addition, Double-Angle and Half-Angle Formulae
The two addition formulae, everything derived from them, and how each derivation works - subtraction, tangent, double angles and half angles in turn.
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The Cartesian Plane, Distance and Midpoint
Ordered pairs, the distance formula as Pythagoras in disguise, and the midpoint as an average of coordinates.
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The Circle and Its Equation
The circle as a locus, its equation from Pythagoras, and recovering centre and radius from an expanded form by completing the square.
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The Dot Product and the Angle Between Vectors
A product of two vectors that returns a number, why it equals |u||v|cos theta, and the test for perpendicularity.
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The Field Laws of Real Number Arithmetic
The commutative, associative, distributive, identity and inverse laws, what each one licenses, and how they justify every step of routine algebraic simplification.
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The Four Forms of a Straight Line
Slope-intercept, point-slope, standard and intercept form - what each is for, and how to move between them.