Algebra 2
Every lesson explains the concept, works through examples, and gives you practice problems with full solutions. Work through them in order, or jump to the topic you need.
43 lessons
- How to Solve Literal Equations and Rearrange FormulasRearrange a formula to solve for any variable using the same inverse operations you use on numbers, with worked examples from geometry, physics, and finance.
- How to Solve Compound Inequalities (AND and OR)Solve compound inequalities joined by AND or OR, graph the solution on a number line, and tell which combinations give no solution or every number.
- How to Solve Absolute Value EquationsSolve absolute value equations by splitting into two cases, why two answers appear, when there is no solution, and how absolute value inequalities differ.
- How to Solve Absolute Value InequalitiesSolve absolute value inequalities by thinking in distances: less than gives one stretch of the number line, greater than gives two rays pointing away.
- Solving Quadratic Equations by FactoringSolve quadratics by factoring using the zero product property, why the equation must equal zero first, and how to factor the common trinomial patterns.
- Completing the Square, Step by StepLearn completing the square: why the method works geometrically, how to handle a leading coefficient, and how it produces vertex form and the quadratic formula.
- The Quadratic Formula: How and Why It WorksLearn the quadratic formula, where it comes from, and how to use it to solve any quadratic equation, with worked examples and practice problems.
- Linear Programming: Optimizing with ConstraintsFind the best value of a linear objective over a region cut out by linear inequalities, and see why the maximum and minimum always sit at a corner.
- Systems of Three Equations in Three VariablesSolve linear systems in x, y and z by elimination: reduce to two equations in two unknowns, solve, substitute back, and see why a solution is where three planes meet.
- Solving Quadratic InequalitiesSolve inequalities like x² − x − 6 > 0 by finding where the parabola crosses the x-axis, testing the intervals between, and writing the answer as intervals.
- Complex Numbers: Arithmetic with iWhy mathematicians introduced i with i² = −1, how to add, subtract and multiply complex numbers, how the powers of i cycle, and how the conjugate makes division possible.
- Quadratic Equations with Complex SolutionsHow the quadratic formula gives complex solutions when the discriminant is negative, why they come in conjugate pairs, and what the Fundamental Theorem of Algebra says.
- Factoring Polynomials: Cubes, Grouping and Quadratic FormFactor polynomials beyond quadratics: sums and differences of cubes, factoring by grouping, and polynomials in quadratic form, then use the factors to solve equations.
- Dividing Polynomials: Long Division and Synthetic DivisionHow polynomial long division mirrors division of numbers, how to write a quotient as q(x) + r(x)/b(x), and why synthetic division is a shortcut for dividing by x − a.
- The Remainder Theorem and the Factor TheoremWhy dividing p(x) by x − a leaves the remainder p(a), why p(a) = 0 means x − a is a factor, and how the two theorems turn one known root into a full factorization.
- The Rational Root Theorem: Finding Zeros of PolynomialsList the possible rational zeros of a polynomial, test them with synthetic division, and use each zero you find to factor and solve the polynomial completely.
- The Fundamental Theorem of AlgebraA polynomial of degree n has exactly n complex zeros, counted with multiplicity. Use that count and conjugate pairs to find every zero and build polynomials.
- Graphing Polynomial Functions: Zeros, Multiplicity and End BehaviorHow to sketch a polynomial from its factored form: where it crosses the x-axis, why a repeated factor makes it bounce, and how the leading term decides both ends.
- Even and Odd Functions: Symmetry of GraphsTest whether a function is even, odd or neither by computing f(−x), and see the answer in its graph: a mirror image in the y-axis or a half turn about the origin.
- Polynomial Identities and the Binomial TheoremWhat makes an equation an identity, how to prove one, how identities generate Pythagorean triples and mental-math shortcuts, and how Pascal's triangle expands (x + y)ⁿ.
- Rational Expressions: Simplifying, Multiplying, AddingHow rational expressions follow the rules of fractions, why only common factors cancel, how to multiply and divide them, and how to add them over a common denominator.
- Direct, Inverse and Joint VariationModel quantities that vary together: direct variation y = kx, inverse variation y = k/x, joint and combined variation, and finding the constant of variation.
- Graphing Rational Functions: Asymptotes, Holes and ZerosHow factoring a rational function reveals its zeros, holes and vertical asymptotes, why the graph shoots off near an asymptote, and how degrees decide the end behavior.
- Simplifying Radical ExpressionsSimplify square roots and cube roots, multiply and divide radicals, add like radicals, and rationalize denominators, including with conjugates.
- Graphing Square Root and Cube Root FunctionsThe shapes of y = √x and y = ∛x, why one starts at a point and the other runs both ways, and how shifts and stretches move the starting point, center, domain and range.
- Radical and Rational Equations, and Extraneous SolutionsWhy every solving step is a claim that follows from the one before, why squaring or clearing denominators can add false answers, and how to solve and check both kinds.
- Polynomial and Rational Inequalities: Sign ChartsSolve polynomial and rational inequalities with a sign chart: find every critical number, test each interval, and handle zeros of the denominator correctly.
- Combining Functions: Sums, Quotients and CompositionHow to build a model by adding, subtracting or dividing functions you know, why adding functions adds their graphs, and how composition feeds one function into another.
- Compound Interest and the Number eInterest compounded n times a year, what happens as n grows without bound, the number e ≈ 2.718, and continuous growth with A = Pe^(rt).
- Logarithms: Solving for an ExponentWhat a logarithm is, why it is the inverse of an exponential, how to use logs to solve for an unknown exponent, and how to evaluate any logarithm with a calculator.
- Properties of Logarithms: Product, Quotient and Power RulesThe three logarithm rules for products, quotients and powers, why they follow from the exponent rules, expanding and condensing logs, and change of base.
- Solving Exponential and Logarithmic EquationsSolve equations with the unknown in an exponent or inside a logarithm: matching bases, taking logs of both sides, condensing logs, and rejecting extraneous solutions.
- Graphs of Exponential and Logarithmic FunctionsGraph y = bˣ and y = log_b x: growth and decay, asymptotes, domain and range, the mirror image in y = x, and shifts and stretches of both.
- Exponential and Logarithmic Models: Half-Life and Log ScalesModel decay, growth and cooling with exponential functions, solve for the time with logarithms, and compare values on pH, decibel and earthquake scales.
- Arithmetic Series and Sigma NotationAdd the terms of an arithmetic sequence with Sₙ = n(a₁ + aₙ)/2, write sums compactly in sigma notation, and use the formula for counting and money problems.
- Geometric Series: The Sum Formula and Loan PaymentsHow to derive the sum of a finite geometric series by multiplying by the ratio and subtracting, and how the same sum sets a savings total and a monthly loan payment.
- Infinite Geometric Series: When an Endless Sum Has a TotalAdd all the terms of a geometric sequence: when −1 < r < 1 the sum is a/(1 − r). Use it to turn repeating decimals into fractions and total a bouncing ball's path.
- The Unit Circle: Sine and Cosine for Every AngleWhy the right-triangle definition stops at 90°, how the unit circle replaces it, the exact values at π/6, π/4 and π/3, and why sine and cosine repeat.
- Graphing Sine and Cosine: Amplitude, Period and MidlineHow unrolling the unit circle produces a wave, what amplitude, period, midline and phase shift each control, and how to fit a sine model to a repeating situation.
- Graphing Tangent and the Reciprocal Trig FunctionsGraph y = tan x with its period π and vertical asymptotes, and the reciprocal functions secant, cosecant and cotangent built from sine and cosine.
- The Pythagorean Identity: sin²θ + cos²θ = 1Why the Pythagorean theorem on the unit circle gives sin²θ + cos²θ = 1, what makes it an identity, and how to find sine, cosine or tangent from one value and a quadrant.
- Solving Trigonometric EquationsSolve equations like 2 sin x = 1: find one angle, use the unit circle to find every solution in one cycle, and add multiples of the period to get them all.
- The Binomial Distribution: Exactly k Successes in n TrialsFind the probability of exactly k successes in n independent trials with the binomial formula, see why it works, and answer at least and at most questions.