𝑥 Algebra

100 topics

• Variables and Expressions (Beginner) – Letters stand for unknown or changing numbers in expressions.
• Evaluating Expressions (Beginner) – Substitute values for variables and simplify.
• Like Terms (Beginner) – Terms with the same variable part can be combined.
• Distributive Property (Beginner) – Multiplication spreads over addition.
• Simplifying Expressions (Beginner) – Combine like terms and apply distribution to write expressions compactly.
• Writing Expressions from Words (Beginner) – Translate phrases like “three more than twice a number” into algebra.
• One-Step Equations (Beginner) – Undo one operation to isolate the variable.
• Two-Step Equations (Beginner) – Undo addition/subtraction, then multiplication/division.
• Multi-Step Equations (Intermediate) – Distribute, combine like terms, move variables to one side.
• Variables on Both Sides (Intermediate) – Collect variable terms on one side first.
• Literal Equations (Intermediate) – Rearrange formulas to solve for a chosen variable.
• Identity and No-Solution Equations (Intermediate) – Some equations are always true or never true.
• Linear Inequalities (Intermediate) – Solve like equations but reverse the sign when multiplying by a negative.
• Compound Inequalities (Intermediate) – Combine two inequalities using “and” or “or”.
• Absolute Value Equations (Intermediate) – Split into two cases: inside equals ± the value.
• Absolute Value Inequalities (Advanced) – Rewrite as a compound inequality.
• Coordinate Plane (Beginner) – Locate points with ordered pairs (x, y).
• Slope (Intermediate) – Rate of change: rise over run.
• Slope-Intercept Form (Intermediate) – Line with slope m and y-intercept b.
• Point-Slope Form (Intermediate) – Line through (x₁,y₁) with slope m.
• Standard Form of a Line (Intermediate) – Ax + By = C describes any line.
• Parallel and Perpendicular Lines (Intermediate) – Parallel lines share slope; perpendicular slopes are negative reciprocals.
• Graphing Linear Equations (Intermediate) – Plot intercepts or two points and draw the line.
• Linear Functions (Intermediate) – Functions with a constant rate of change.
• Function Notation (Intermediate) – f(x) names the output for input x.
• Domain and Range (Intermediate) – Domain is all allowed inputs; range is all outputs.
• Relations vs Functions (Intermediate) – A function gives exactly one output per input (vertical line test).
• Systems of Equations: Graphing (Intermediate) – Solution is where two lines intersect.
• Systems: Substitution (Intermediate) – Solve one equation for a variable and substitute into the other.
• Systems: Elimination (Intermediate) – Add or subtract equations to cancel a variable.
• Systems with Three Variables (Advanced) – Reduce to two variables then solve in sequence.
• Systems of Inequalities (Advanced) – Solution region is the overlap of half-planes.
• Linear Programming (Expert) – Optimise a linear objective over a feasible polygon; maxima occur at vertices.
• Exponent Rules for Variables (Intermediate) – Apply laws of exponents to algebraic terms.
• Polynomials: Terms and Degree (Intermediate) – Sums of terms cxⁿ; the degree is the highest power.
• Adding and Subtracting Polynomials (Intermediate) – Combine like terms.
• Multiplying Polynomials (Intermediate) – Distribute every term of one factor across the other.
• FOIL Method (Intermediate) – Multiply two binomials: First, Outer, Inner, Last.
• Special Products (Intermediate) – Perfect squares and difference of squares patterns.
• Factoring Out the GCF (Intermediate) – Pull the greatest common factor out of every term.
• Factoring Trinomials (Intermediate) – Find two numbers that multiply to c and add to b.
• Factoring by Grouping (Advanced) – Group terms in pairs to find a common binomial.
• Difference of Squares (Intermediate) – A square minus a square factors into conjugates.
• Sum and Difference of Cubes (Advanced) – Special factoring patterns for cubes.
• Quadratic Equations by Factoring (Intermediate) – Set to zero, factor, use the zero-product property.
• Completing the Square (Advanced) – Rewrite a quadratic as a perfect square plus constant.
• Quadratic Formula (Advanced) – Gives roots of any quadratic.
• Discriminant (Advanced) – b²−4ac tells the number and type of roots.
• Parabolas and Vertex Form (Advanced) – Graph of a quadratic; vertex at (h,k).
• Quadratic Axis of Symmetry (Advanced) – Parabola symmetric about x = −b/2a.
• Quadratic Inequalities (Advanced) – Find where a parabola is above or below the axis.
• Quadratic Applications (Advanced) – Projectile motion, area and optimisation problems.
• Rational Expressions (Advanced) – Fractions with polynomials in numerator and denominator.
• Simplifying Rational Expressions (Advanced) – Factor and cancel common factors.
• Rational Expression Operations (Advanced) – Add, multiply and divide like numeric fractions.
• Rational Equations (Advanced) – Multiply by the LCD to clear denominators; check extraneous solutions.
• Rational Functions and Asymptotes (Advanced) – Graphs with vertical and horizontal asymptotes.
• Radical Expressions (Advanced) – Simplify and operate with roots.
• Radical Equations (Advanced) – Isolate the root, square both sides, check solutions.
• Rational Exponents (Advanced) – Link roots and powers.
• Square Root Functions (Advanced) – Graph of y = √x translated and scaled.
• Piecewise Functions (Advanced) – Functions defined by different rules on different intervals.
• Absolute Value Functions (Advanced) – V-shaped graph.
• Function Transformations (Advanced) – Shifts, stretches and reflections of graphs.
• Function Composition (Advanced) – Apply one function after another.
• Inverse Functions (Advanced) – Undo a function; graph reflects across y = x.
• Even and Odd Functions (Advanced) – Even: f(−x)=f(x); odd: f(−x)=−f(x).
• Polynomial Functions and Graphs (Advanced) – End behaviour and turning points depend on degree.
• Polynomial Long Division (Advanced) – Divide polynomials like integers to find quotient and remainder.
• Synthetic Division (Advanced) – A shortcut for dividing by (x − c).
• Remainder and Factor Theorems (Advanced) – p(c) is the remainder when dividing by x−c; zero remainder means a factor.
• Rational Root Theorem (Expert) – Possible rational roots are ± factors of constant over factors of leading coefficient.
• Fundamental Theorem of Algebra (Expert) – Every degree-n polynomial has exactly n complex roots counted with multiplicity.
• Complex Numbers (Advanced) – Numbers a + bi where i² = −1.
• Complex Arithmetic (Advanced) – Add, multiply and divide using i² = −1 and conjugates.
• Complex Conjugates (Advanced) – Reflect across the real axis; product gives a real number.
• Exponential Functions (Advanced) – Functions where the variable is the exponent.
• Exponential Growth and Decay (Advanced) – Constant percent change over equal time steps.
• The Number e (Advanced) – The base where growth rate equals value.
• Logarithms (Advanced) – The inverse of exponentiation.
• Laws of Logarithms (Advanced) – Convert products to sums, powers to multiples.
• Change of Base Formula (Advanced) – Convert logs between bases.
• Exponential and Log Equations (Advanced) – Take logs or exponentiate both sides.
• Natural Logarithm (Advanced) – Log with base e.
• Arithmetic Series (Advanced) – Sum of an arithmetic sequence.
• Geometric Series (Advanced) – Sum of a geometric sequence.
• Infinite Geometric Series (Advanced) – Converges when |r|<1.
• Sigma Notation (Advanced) – Compact notation for sums.
• Binomial Theorem (Expert) – Expand (a+b)ⁿ with binomial coefficients.
• Partial Fractions (Expert) – Decompose a rational function into simpler fractions.
• Mathematical Induction (Expert) – Prove statements for all n by base case and inductive step.
• Matrices (Algebra Intro) (Advanced) – Rectangular arrays used to solve systems.
• Cramer’s Rule (Expert) – Solve linear systems using determinants.
• Conic Sections: Overview (Expert) – Circles, ellipses, parabolas, hyperbolas as plane sections of a cone.
• Parametric Equations (Expert) – Describe curves with x(t) and y(t).
• Polar Coordinates (Algebra View) (Expert) – Points by distance r and angle θ.
• Inequalities: AM–GM (Expert) – The arithmetic mean is at least the geometric mean.
• Cauchy–Schwarz Inequality (Expert) – Bounds dot products.
• Functional Equations (Expert) – Find functions satisfying given relations.
• Algebraic Word Problem Modelling (Intermediate) – Build equations from real situations.