Openstax Elementary Algebra 2E

In Lesson 1.1 of OpenStax Elementary Algebra 2E, students review foundational math concepts including place value with whole numbers, identifying multiples using divisibility tests, and finding prime factorizations and least common multiples. The lesson introduces counting numbers, whole numbers, and the place value system as essential groundwork before advancing into algebraic thinking. This material serves as a prerequisite review for students beginning an elementary algebra course.

In this lesson from OpenStax Elementary Algebra 2E, students learn the foundational language of algebra, including how to use variables, constants, and algebraic symbols to represent quantities and operations. The lesson covers simplifying expressions using the order of operations, evaluating expressions, identifying and combining like terms, and translating English phrases into algebraic expressions. These skills form the essential vocabulary and notation students need to work with algebraic concepts throughout the course.

In this lesson from OpenStax Elementary Algebra 2E, students learn to work with integers by understanding negative numbers, opposites, and absolute value on the number line. The lesson then builds to adding and subtracting integers using number line models and rules for positive and negative values. Suitable for middle school and early high school learners, this section provides the integer foundation needed for algebraic reasoning throughout the course.

Students learn to multiply and divide integers using signed number rules, including how same signs produce a positive result and different signs produce a negative result. The lesson also covers simplifying and evaluating variable expressions with integers, and translating English phrases into algebraic expressions. This content is from Lesson 1.4 of OpenStax Elementary Algebra 2E, Chapter 1: Foundations.

In this lesson from OpenStax Elementary Algebra 2E, students learn core fraction concepts including how to find equivalent fractions, simplify fractions, and multiply and divide fractions. The lesson also covers simplifying expressions written with a fraction bar and translating verbal phrases into fractional expressions. These foundational skills build the algebraic reasoning needed for more advanced equation work throughout the course.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to add and subtract fractions with both common and different denominators, including finding the least common denominator (LCD) by identifying the least common multiple of the denominators. The lesson also covers using the order of operations to simplify complex fractions and evaluating variable expressions with fractions. Part of the foundational chapter in an algebra course, this section builds essential fraction skills needed throughout algebra.

In this lesson from OpenStax Elementary Algebra 2E, students learn to name and write decimals using place value (tenths, hundredths, thousandths), round decimals, and perform addition, subtraction, multiplication, and division with decimal numbers. The lesson also covers converting between decimals, fractions, and percents, reinforcing the connection between decimal notation and fractions with denominators that are powers of ten.

In this lesson from OpenStax Elementary Algebra 2E, students learn to simplify expressions with square roots, identify perfect squares, and understand principal square root notation using the radical sign. The lesson also covers classifying numbers as integers, rational, irrational, or real, and locating fractions and decimals on the number line. These foundational skills support algebra readiness for middle and high school students.

In this lesson from OpenStax Elementary Algebra 2E, students learn to apply the commutative, associative, identity, inverse, and distributive properties of real numbers to simplify algebraic expressions. The lesson covers why order matters in addition and multiplication but not in subtraction or division, and how strategic grouping using the associative property can make calculations easier. These foundational properties prepare students to work fluently with real numbers throughout their algebra studies.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to make unit conversions within the U.S. system and the metric system, use mixed units of measurement, and convert between the two systems as well as between Fahrenheit and Celsius temperatures. The core technique taught is using the identity property of multiplication to set up unit fractions that cancel unwanted units, applied to length, weight, volume, and time. This foundational algebra lesson is suitable for students transitioning from pre-algebra into a first formal algebra course.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to verify solutions of equations and apply the Subtraction and Addition Properties of Equality to solve one-variable linear equations. The lesson covers substituting values to check whether a number is a solution, isolating the variable by adding or subtracting the same quantity on both sides of an equation, and translating real-world problems into equations to solve. It also includes solving equations that require simplification before applying these properties.

In this lesson from OpenStax Elementary Algebra 2E, students learn to solve one-variable linear equations using the Division and Multiplication Properties of Equality, including cases where a variable is multiplied by a constant such as 2x = 6. The lesson also covers solving equations that require simplification and translating real-world application problems into equations. Students practice verifying solutions and build fluency with isolating variables through inverse operations.

In this lesson from OpenStax Elementary Algebra 2E, students learn to solve linear equations that have variables and constants on both sides by applying the Addition, Subtraction, Division, and Multiplication Properties of Equality. The core strategy involves choosing a "variable side" and a "constant side," then isolating the variable to reach the standard form ax = b. Students also practice translating real-world word problems into equations and solving them using these techniques.

In this lesson from OpenStax Elementary Algebra 2E, students learn a five-step general strategy for solving any linear equation, including applying the Distributive Property, combining like terms, and using the Addition, Subtraction, Multiplication, and Division Properties of Equality. The lesson covers equations with constants on both sides, variables on both sides, and combinations of both, as well as how to classify equations. Students practice checking their solutions by substituting back into the original equation to verify a true statement.

In this lesson from OpenStax Elementary Algebra 2E, students learn to solve linear equations with fraction and decimal coefficients using two key techniques: applying the Multiplication Property of Equality to clear fractions by multiplying through by the least common denominator, and eliminating decimals by multiplying by a power of ten. The lesson covers finding the LCD of fraction coefficients, using the distributive property after clearing, and then solving the resulting whole-number equation with the general strategy. Real-world applications, such as solving perimeter and coin problems with decimal coefficients, reinforce both methods.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to apply the distance, rate, and time formula (d = rt) to solve real-world problems and how to isolate a specific variable in a formula by rearranging its terms. Learners practice translating word problems into equations, substituting known values, and solving for an unknown quantity such as distance, rate, or time. This foundational algebra skill prepares students to work fluently with a wide range of mathematical and scientific formulas.

In this lesson from OpenStax Elementary Algebra 2E, students learn to solve linear inequalities using the Subtraction, Addition, Division, and Multiplication Properties of Inequality, including the critical rule that multiplying or dividing by a negative number reverses the inequality symbol. Students also practice graphing solution sets on the number line using open and closed parentheses and brackets, and expressing solutions in interval notation. The lesson covers inequalities that require multi-step simplification and translating real-world situations into inequalities to solve.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to apply a structured problem-solving strategy to translate word problems into algebraic equations and solve them. The strategy walks through steps including identifying what is being asked, assigning a variable, writing an equation, and checking the solution. The lesson also covers solving basic number problems using linear equations.

In this lesson from OpenStax Elementary Algebra 2E, students learn to translate and solve basic percent equations, applying algebraic methods to real-world problems involving tips, sales tax, discounts, simple interest, and markups. Key skills include finding an unknown number, an unknown percent, and calculating percent increase and percent decrease using a structured problem-solving strategy. The content is designed for elementary algebra students building fluency with decimal conversions and equation solving.

In this lesson from OpenStax Elementary Algebra 2E, students learn to solve mixture applications by setting up and solving linear equations for coin problems, ticket and stamp problems, and investment problems involving simple interest. Using the mixture model — where number times value equals total value — students practice organizing information in tables and writing algebraic expressions to find unknown quantities. The lesson connects real-world contexts like counting coins and calculating interest to core algebra skills in equation setup and problem-solving strategy.

Students learn to solve geometry applications involving triangles, rectangles, and the Pythagorean Theorem using properties such as the triangle angle sum (m∠A + m∠B + m∠C = 180°), perimeter and area formulas, and right triangle relationships. This lesson is part of Chapter 3: Math Models in the OpenStax Elementary Algebra 2E textbook and covers setting up and solving algebraic equations from real-world geometric scenarios. Students practice a structured problem-solving strategy to find unknown side lengths, angles, and areas across multiple shape types.

In this lesson from Openstax Elementary Algebra 2E, students learn to solve uniform motion problems using the distance, rate, and time formula D = rt. Learners apply a structured problem-solving strategy that includes drawing diagrams and organizing rate, time, and distance data in tables to set up and solve algebraic equations involving two travel scenarios. This lesson builds key skills in translating real-world motion contexts into linear equations and interpreting solutions in context.

In this lesson from OpenStax Elementary Algebra 2E, Chapter 3, students learn how to solve real-world applications using linear inequalities by translating written scenarios into algebraic inequalities and solving for an unknown variable. Using a structured step-by-step method, students work through problems involving income thresholds, weight limits, and budget constraints to find maximum or minimum values. The lesson also addresses cases where solutions must be rounded to whole numbers based on the context of the problem.

Students learn to plot ordered pairs in the rectangular coordinate system by identifying x-coordinates and y-coordinates on the x-axis and y-axis, and locating points across the four quadrants. The lesson also covers verifying solutions to equations in two variables and completing tables of solutions to linear equations. This is from Chapter 4 of OpenStax Elementary Algebra 2E, which introduces the coordinate plane as the foundation for graphing linear equations.

In this lesson from OpenStax Elementary Algebra 2E, students learn to graph linear equations in two variables by plotting ordered pairs as points in the rectangular coordinate system and connecting them to form a line. Students explore the relationship between solutions of an equation like 3x + 2y = 6 and its graph, understanding that every point on the line is a solution and every solution corresponds to a point on the line. The lesson also covers graphing special cases, including vertical and horizontal lines.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to identify x-intercepts and y-intercepts on a graph, find intercepts algebraically from a linear equation, and use those two points to graph a line. The lesson covers the definitions of the x-intercept as the point (a, 0) where a line crosses the x-axis and the y-intercept as the point (0, b) where it crosses the y-axis. This foundational algebra skill gives students an efficient alternative to plotting multiple points when graphing linear equations.

In this lesson from OpenStax Elementary Algebra 2E, students learn to understand and calculate the slope of a line using the rise-over-run formula (m = rise/run). Students explore slope through geoboard models, graphs, and the slope formula to find the slope between two points, and also examine the slope of horizontal and vertical lines. Real-world applications such as roof pitch, highway grades, and wheelchair ramps are used to connect the concept of slope to everyday contexts.

In this lesson from OpenStax Elementary Algebra 2E, students learn to identify the slope and y-intercept directly from an equation written in slope-intercept form (y = mx + b) and use those values to graph a line. The lesson also covers how to use slopes to identify parallel and perpendicular lines, and how to choose the most convenient method for graphing a linear equation. Suitable for high school or college-level introductory algebra, this section builds on prior work with graphing linear equations in two variables.

In this lesson from OpenStax Elementary Algebra 2E, Chapter 4, students learn how to find the equation of a line using slope-intercept form (y = mx + b) and point-slope form (y − y₁ = m(x − x₁)). Students practice writing linear equations given the slope and y-intercept, the slope and a single point, or two points, as well as finding equations for parallel and perpendicular lines. The lesson connects these algebraic skills to real-world linear modeling in science, economics, and engineering.

In this lesson from OpenStax Elementary Algebra 2E, students learn to graph linear inequalities in two variables by verifying whether ordered pairs are solutions, identifying boundary lines of the form Ax + By = C, and shading the correct half-plane to represent all solutions. The lesson covers how to determine whether a boundary line is solid or dashed based on whether the inequality is strict or includes equality. Students build on prior knowledge of solving one-variable inequalities and graphing linear equations to interpret solution regions on the coordinate plane.

Students learn how to solve systems of linear equations by graphing, including how to determine whether an ordered pair is a solution to a system and how to identify the number of solutions a linear system has. This lesson from Chapter 5 of Openstax Elementary Algebra 2E covers substituting ordered pairs to verify solutions and finding the intersection point of two lines on a coordinate plane. It provides the foundation for later methods such as substitution and elimination.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to solve a system of linear equations using the substitution method, which involves isolating one variable in an equation and substituting that expression into the other equation to find an exact solution. The lesson walks through a step-by-step process of solving for one variable, substituting, simplifying, and checking the solution in the original system. Students also apply the substitution method to real-world word problems where graphing would be imprecise or impractical.

Students learn how to solve systems of linear equations using the Elimination Method, which involves adding equations together to cancel out one variable by making its coefficients opposites. The lesson, from Chapter 5 of Openstax Elementary Algebra 2E, also covers solving real-world applications such as calorie-burning and rate-of-travel problems using this technique. Students additionally practice choosing the most convenient method among graphing, substitution, and elimination for a given system.

In this lesson from OpenStax Elementary Algebra 2E, Chapter 5, students learn how to translate real-world word problems into systems of linear equations and solve them using substitution or elimination. Applications include direct translation problems, geometry scenarios, and uniform motion problems involving rates and distances. Students also practice choosing the most convenient solution method based on the structure of each system.

In this lesson from OpenStax Elementary Algebra 2E, students learn to solve mixture applications and interest problems by setting up and solving systems of two linear equations with two variables. Using organized tables, learners translate real-world scenarios involving tickets, coins, and percent interest into systems and apply methods such as elimination and substitution to find solutions. The lesson builds on earlier single-variable mixture strategies and extends them to a two-variable approach covered in Chapter 5.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to determine whether an ordered pair is a solution to a system of linear inequalities by substituting values into each inequality. Students then practice solving systems of linear inequalities by graphing, identifying the overlapping shaded region on the coordinate plane that satisfies all inequalities simultaneously. The lesson also covers real-world applications of systems of inequalities, making it a natural extension of earlier work with systems of linear equations in Chapter 5.

In this lesson from OpenStax Elementary Algebra 2E, students learn to identify and classify polynomials as monomials, binomials, or trinomials, determine the degree of polynomial expressions, and perform addition and subtraction of polynomials by combining like terms. Students also practice evaluating a polynomial for a given value. The lesson provides the foundational vocabulary and skills needed for the polynomial operations covered throughout Chapter 6.

In this lesson from OpenStax Elementary Algebra 2E, students learn to simplify expressions with exponents by applying the Product Property, Power Property, and Product to a Power Property for exponents. The lesson covers exponential notation, the difference between expressions like (-5)^4 and -5^4, and how to multiply monomials using these properties. Students practice evaluating and simplifying both numeric and variable expressions involving repeated multiplication of factors.

In OpenStax Elementary Algebra 2E, Chapter 6, Lesson 3, students learn how to multiply polynomials using the Distributive Property, covering three cases: multiplying a polynomial by a monomial, multiplying a binomial by a binomial, and multiplying a trinomial by a binomial. The lesson introduces the FOIL method and vertical multiplication as additional strategies for binomial multiplication, with practice simplifying results by combining like terms. This foundational algebra skill builds directly on students' prior knowledge of the Distributive Property and exponent rules.

In this lesson from OpenStax Elementary Algebra 2E, students learn to apply the Binomial Squares Pattern and the Product of Conjugates Pattern to multiply special polynomial expressions efficiently. The lesson covers squaring a binomial using the formula (a+b)² = a² + 2ab + b² and multiplying conjugate pairs to produce a difference of squares. Students also practice recognizing which special product pattern applies to a given expression, building fluency with polynomial multiplication techniques used throughout algebra.

In this lesson from OpenStax Elementary Algebra 2e, Chapter 6, students learn to divide monomials by applying the Quotient Property for Exponents, which involves subtracting exponents when dividing expressions with the same base. The lesson also covers simplifying expressions with zero exponents and using the Quotient to a Power Property, building on earlier multiplication exponent rules. Students practice combining multiple exponent properties to fully simplify algebraic expressions involving monomial division.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to divide a polynomial by a monomial and divide a polynomial by a binomial using fraction addition properties and long division. Students practice splitting polynomial expressions into separate terms, simplifying quotients with variables and exponents, and carefully managing signs when dividing by negative monomials. This foundational algebra skill builds directly on prior knowledge of monomial division and prepares students for more advanced polynomial operations.

In this lesson from OpenStax Elementary Algebra 2E, students learn to work with integer exponents, including the definition of negative exponents (a⁻ⁿ = 1/aⁿ) and how to simplify expressions using the Quotient Property for Exponents. The lesson also covers converting numbers between standard decimal notation and scientific notation, as well as multiplying and dividing values written in scientific notation. These skills build on earlier polynomial and exponent concepts from Chapter 6 and are applicable to real-world contexts involving very large or very small quantities.

This lesson from OpenStax Elementary Algebra 2e introduces students to factoring polynomials, beginning with how to find the greatest common factor (GCF) of two or more expressions by writing coefficients as prime factors and identifying shared variables. Students then apply the GCF to factor it out of a polynomial and use the factor by grouping method to factor expressions with four terms. These foundational factoring skills prepare students for solving quadratic equations and working with more complex polynomial expressions throughout Chapter 7.

In this lesson from OpenStax Elementary Algebra 2E, Chapter 7, students learn to factor trinomials of the form x²+bx+c by reversing the FOIL multiplication process to find two binomial factors (x+m)(x+n). The core skill involves identifying factor pairs of the constant term c whose sum equals the middle coefficient b, and the lesson extends this to two-variable trinomials of the form x²+bxy+cy². Practice problems include mixed factoring, real-world applications such as modeling the height of a baseball and the area of a rectangle, and exercises on finding the greatest common factor and factoring by grouping.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to factor trinomials of the form ax²+bx+c using three approaches: factoring out the greatest common factor (GCF) first, trial and error, and the "ac" method. The lesson also introduces a preliminary factoring strategy that helps students identify which method to apply based on the number of terms and structure of a polynomial. This material is typically covered in a high school or college-level introductory algebra course as part of a broader unit on factoring polynomials completely.

In this lesson from OpenStax Elementary Algebra 2E, students learn to factor special products including perfect square trinomials, differences of squares, and sums and differences of cubes by recognizing patterns such as a² + 2ab + b² = (a+b)² and a² − b² = (a+b)(a−b). Students practice identifying when a polynomial fits one of these special forms and applying the corresponding pattern to factor it efficiently. The lesson also guides students in choosing the most appropriate factoring method to factor a polynomial completely.

In this lesson from OpenStax Elementary Algebra 2E, Chapter 7, students learn a general step-by-step strategy for factoring polynomials completely by identifying the appropriate method based on polynomial type. The lesson covers factoring out the greatest common factor first, then applying techniques such as difference of squares, sum and difference of cubes, perfect square trinomials, the ac method, and factoring by grouping. Practice problems range from straightforward binomials and trinomials to applied real-world contexts involving area expressions.

In this lesson from OpenStax Elementary Algebra 2E, students learn to solve quadratic equations of the form ax² + bx + c = 0 using the Zero Product Property and factoring techniques. The lesson covers how to set each factor equal to zero, solve the resulting linear equations, and check solutions. Real-world applications, such as modeling the height of a falling object over time, are used to reinforce how quadratic equations appear in everyday contexts.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to work with rational expressions — fractions where the numerator and denominator are polynomials — covering how to determine values that make a rational expression undefined, evaluate rational expressions, and simplify them including cases with opposite factors. Students practice setting the denominator equal to zero to find excluded values and apply polynomial factoring skills to reduce rational expressions to simplest form.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to multiply and divide rational expressions by applying the same principles used with numerical fractions, including multiplying numerators and denominators and simplifying by removing common factors. The lesson covers factoring polynomials in the numerator and denominator to identify and cancel shared factors before or after multiplying, and extends to dividing rational expressions by multiplying by the reciprocal. Real-world applications, such as calculating average tax rates and combined work times using rational formulas, reinforce these algebraic skills in practical contexts.

In this lesson from OpenStax Elementary Algebra 2E, Chapter 8, students learn how to add and subtract rational expressions that share a common denominator by combining numerators over the shared denominator and simplifying the result. The lesson also covers subtracting rational expressions and working with expressions whose denominators are opposites. Real-world applications, such as probability problems, reinforce how these algebraic techniques connect to practical contexts.

In this algebra lesson from OpenStax Elementary Algebra 2E, students learn how to add and subtract rational expressions with unlike denominators by finding the least common denominator (LCD) through factoring polynomial expressions. The lesson covers identifying equivalent rational expressions, rewriting fractions with a shared LCD, and performing addition and subtraction across expressions with different polynomial denominators. This material builds on foundational fraction skills and polynomial factoring, making it appropriate for high school algebra students.

In this lesson from OpenStax Elementary Algebra 2E, students learn to simplify complex rational expressions — rational expressions that contain fractions in their numerator or denominator — using two methods: rewriting as division and multiplying by the reciprocal, or multiplying through by the least common denominator (LCD). Students practice applying these techniques to expressions with variables in denominators, including cases requiring factoring to identify the LCD. Real-world problems involving rates, such as shared work and river current scenarios, reinforce how these skills apply in context.

In this OpenStax Elementary Algebra 2E lesson, students learn how to solve rational equations by finding the least common denominator and multiplying both sides to clear fractions. The lesson also covers identifying and discarding extraneous solutions — values that make a denominator equal to zero and render the equation undefined. Students practice solving for a specific variable within rational equations, building on prior skills with linear equations and rational expressions.

In this lesson from OpenStax Elementary Algebra 2E, students learn to solve proportions by setting up equations of the form a/b = c/d and using the LCD to clear fractions and isolate the variable. The lesson also covers similar figure applications, connecting proportional reasoning to real-world scaling problems. Students practice checking solutions by substituting back into the original proportion to verify accuracy.

In this lesson from OpenStax Elementary Algebra 2E, students learn to solve uniform motion applications using the formula D = rt and its rearranged form t = D/r, setting up rational equations to find unknown speeds involving headwinds, tailwinds, and combined travel scenarios. Students also solve work applications, where two workers or machines complete a job together, using rational expressions to model each contributor's rate. The lesson builds on solving rational equations with variables in the denominator, applying cross-multiplication and LCD methods to real-world problems.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to solve direct variation and inverse variation problems by identifying the constant of variation and writing equations of the form y = kx or y = k/x. Learners practice setting up and solving proportional relationships using real-world contexts such as hourly wages and calorie burning. The lesson builds on skills with proportions and algebraic equation solving covered earlier in Chapter 8.

In this lesson from OpenStax Elementary Algebra 2E, students learn to simplify expressions with square roots, including identifying principal square roots, negative square roots, and recognizing when a square root is not a real number. The lesson also covers estimating and approximating square roots and simplifying variable expressions with square roots. Students work with radical sign notation and perfect squares as an introduction to Chapter 9's broader study of roots and radicals.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to simplify square roots using the Product Property and Quotient Property of square roots, including removing perfect square factors from a radicand. The lesson also covers simplifying variable expressions with square roots involving terms like the square root of q squared or expressions with multiple variables and even exponents. Practice problems range from straightforward radical simplification to real-world applications involving area and side length.

This lesson from OpenStax Elementary Algebra 2E teaches students how to add and subtract like square roots by combining coefficients when radicands match, just as they would combine like terms in algebra. Students also learn to simplify square roots using the Product and Quotient Properties before adding or subtracting expressions with unlike radicands. Real-world applications, such as finding garden dimensions and calculating falling-object times, reinforce these radical arithmetic skills.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to multiply square roots using the Product Property of Square Roots, which states that the square root of a product equals the product of the square roots. Students practice multiplying radicals with coefficients and variables, then simplifying the resulting expressions by identifying and removing perfect square factors. The lesson also prepares students to apply polynomial multiplication techniques to expressions involving square roots.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to divide square roots by applying the Quotient Property of Square Roots, simplifying radical expressions in fraction form, and working with variable radicands using the Quotient Property for Exponents. The lesson also covers rationalizing one-term and two-term denominators to eliminate radicals from the denominator of a fraction. Suitable for college-level developmental math or advanced high school algebra courses, this section builds essential skills for simplifying complex radical expressions.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to solve radical equations by isolating the square root and squaring both sides to eliminate the radical. The lesson covers identifying and discarding extraneous solutions that arise from the squaring process, as well as applying square root equations to real-world problems such as calculating fall time from altitude.

In this lesson from OpenStax Elementary Algebra 2E, Chapter 9, students learn to work with higher roots, including cube roots, fourth roots, and fifth roots, extending their understanding beyond square roots. Students practice simplifying radical expressions using the nth root definition, the Product Property, and the Quotient Property, as well as adding and subtracting higher roots. The lesson also introduces the concept of the index of a radical and explains why even roots of negative numbers are not real numbers.

In this lesson from OpenStax Elementary Algebra 2E, students learn to simplify expressions with rational exponents, including forms such as a^(1/n) and a^(m/n), and connect these to equivalent radical notation. Students apply the Laws of Exponents — including the Power Property — to rewrite and simplify expressions involving fractional exponents. The lesson builds fluency in moving between radical and rational exponent forms to solve algebraic problems efficiently.

In this lesson from OpenStax Elementary Algebra 2E, Chapter 10, students learn to solve quadratic equations using the Square Root Property, applying it to equations of the form ax² = k and a(x−h)² = k. Students practice isolating the quadratic term, applying the property to produce both positive and negative solutions expressed as ±√k, and simplifying radical expressions including non-perfect squares. The lesson builds on prior knowledge of factoring and square roots to introduce a more versatile method for solving quadratic equations.

In this lesson from OpenStax Elementary Algebra 2E, students learn how to solve quadratic equations by completing the square, a method that transforms expressions like x² + bx into perfect square trinomials by adding (½b)². The lesson covers completing the square for binomial expressions and applying the technique to solve quadratic equations in both the forms x² + bx + c = 0 and ax² + bx + c = 0 using the Square Root Property.

Students learn to solve quadratic equations using the quadratic formula, x = (−b ± √(b²−4ac)) / 2a, derived by completing the square on the general standard form ax²+bx+c=0. The lesson also covers using the discriminant (b²−4ac) to predict the number of solutions and identifying the most appropriate method for solving a given quadratic equation. This material is part of Chapter 10 of OpenStax Elementary Algebra 2E.

In this lesson from OpenStax Elementary Algebra 2E, students learn to solve real-world applications modeled by quadratic equations, applying the quadratic formula and factoring to problems involving consecutive odd and even integers, geometry, and the Pythagorean theorem. Students practice translating word problems into standard form quadratic equations and interpreting solutions in context. The lesson builds problem-solving skills by identifying the most appropriate method — factoring, completing the square, or the quadratic formula — for each application.

In this lesson from OpenStax Elementary Algebra 2E, students learn to graph quadratic equations in two variables of the form y = ax² + bx + c by plotting points and recognizing the resulting U-shaped curve as a parabola. Key skills include identifying the axis of symmetry, finding the vertex and intercepts of a parabola, and determining whether it opens upward or downward based on the sign of the leading coefficient. Students also apply these graphing techniques to solve maximum and minimum value problems.