Completing the Square and Vertex Form, Step by Step
Learn how to complete the square to write a quadratic in vertex form a(x − h)² + k, find the vertex, and see why it matters for graphs.
Algebra
15 guides.
Complex Numbers for Beginners: i, Arithmetic and the Argand Diagram
What i = √−1 means, how to add, multiply and divide complex numbers, and how modulus, argument and the Argand diagram work.
Exponential Growth and Decay: Formula, Doubling Time and Half-Life
Understand exponential growth and decay, use A = A₀(1 + r)ᵗ, and find doubling times and half-lives with logarithms. Worked examples.
How to Factor Polynomials: A Practical Checklist
A step-by-step checklist for factoring polynomials: common factors, differences of squares, quadratics, grouping and the factor theorem.
How to Find the Determinant of a Matrix
Calculate 2×2 and 3×3 determinants with ad − bc and cofactor expansion, learn what a zero determinant means, and see it as an area.
How to Find the Inverse of a Matrix (2×2 and 3×3)
Find the inverse of a 2×2 matrix with the quick formula and of a 3×3 matrix by Gauss–Jordan elimination, and check it with A × A⁻¹ = I.
How to Solve Cubic Equations Using the Factor Theorem
Solve cubic equations by finding a root with the rational root theorem, dividing it out, and solving the remaining quadratic. Worked examples.
How to Solve Linear Inequalities (and When to Flip the Sign)
Solve linear inequalities step by step, learn the one rule that flips the sign, and show answers on a number line and in interval notation.
Vectors Explained: Magnitude, Dot Product and Cross Product
Learn what vectors are, how to find magnitude, the dot product and the angle between vectors, and how the 3D cross product works.
Arithmetic vs Geometric Sequences: Formulas and Examples
Tell arithmetic and geometric sequences apart, find the nth term and the sum of each, and see when a geometric series has a sum to infinity.
How to Simplify Square Roots (Surds) Step by Step
Simplify square roots like √50 and √72 by taking out square factors, combine surds, and rationalise denominators, with clear worked examples.
How to Solve Simultaneous Equations: Substitution vs Elimination
Solve pairs of simultaneous equations by substitution or elimination, see what the answer means on a graph, and spot when there is no solution.
Logarithms Explained for Beginners
What a logarithm is, how it undoes an exponent, the three log laws, the change-of-base formula and how to solve exponential equations.
Three Ways to Solve a Quadratic Equation
Solve quadratic equations by factoring, the quadratic formula and completing the square, with worked examples and tips on choosing a method.
What Is the Discriminant? Reading a Quadratic at a Glance
The discriminant b² − 4ac tells you how many roots a quadratic has and what kind they are. Learn to use it with clear examples.