OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials

Free Algebra Study Guide with Videos

Welcome to OpenAlgebra.com, your comprehensive online destination for mastering algebra. Dive deep into our extensive collection of step-by-step tutorials, complemented by illustrative YouTube videos for a visual learning experience. Whether you're a student seeking supplemental material beyond your textbooks or a teacher looking for a rich resource of worked-out math problems, our platform offers a treasure trove of algebraic knowledge tailored to enhance your understanding.

Chapter 1 - Real Numbers and Their Operations

1.1 Real Numbers and The Number Line
1.2 Adding and Subtracting Integers
1.3 Multiplying and Dividing Integers
1.4 Fractions
1.5 Review of Decimals and Percents
1.6 Exponents and Square Roots
1.7 Order of Operations
1.8 Sample Exam Questions

Chapter 2 - Linear Equations and Inequalities

2.1 Introduction to Algebra
2.2 Simplifying Algebraic Expressions
2.3 Linear Equations: Part I
2.4 Linear Equations: Part II
2.5 Applications of Linear Equations
2.6 Ratio and Proportion Applications
2.7 Introduction to Inequalities and Interval Notation
2.8 Linear Inequalities (one variable)
2.9 Review Exercises and Sample Exam

Chapter 3 - Graphing Lines

3.1 Rectangular Coordinate System
3.2 Graph by Plotting Points
3.3 Graph using Intercepts
3.4 Graph using the y-intercept and Slope
3.5 Finding Linear Equations
3.6 Parallel and Perpendicular Lines
3.7 Introduction to Functions
3.8 Linear Inequalities (Two Variables)
3.9 Review Exercises and Sample Exam

Chapter 4 - Solving Linear Systems

4.1 Solving Linear Systems by Graphing
4.2 Solving Linear Systems by Substitution
4.3 Solving Linear Systems by Elimination
4.4 Applications of Linear Systems
4.5 Solving Systems of Linear Inequalities (Two Variables)
4.6 Review Exercises and Sample Exam

[ Elementary Algebra Exam #1 Solutions ]
[ Elementary Algebra Exam #2 Solutions ]
[ Elementary Algebra Exam #3 Solutions ]
[ Elementary Algebra Exam #4 Solutions ]

--- EA Sample Final Exam Ch. 1-7 ---

Chapter 5 - Polynomials and Their Operations

5.1 Rules of Exponents (Integer Exponents)
5.2 Introduction to Polynomials and Evaluating
5.3 Adding and Subtracting Polynomials
5.4 Multiplying Polynomials and Special Products
5.5 Dividing Polynomials (Synthetic Division)
5.6 Negative Exponents and Scientific Notation
5.7 Review Exercises and Sample Exam

Chapter 6 - Factoring and Solving by Factoring

6.1 Introduction to Factoring
6.2 Factoring Trinomials
6.3 Factoring Trinomials ax^2 + bx + c
6.4 Factoring Binomials
6.5 General Guidelines for Factoring Polynomials
6.6 Solving Equations by Factoring
6.7 Applications involving Quadratic Equations
6.8 Review Exercises and Sample Exam

Chapter 7 - Rational Expressions and Equations

7.1 Simplifying Rational Expressions
7.2 Multiplying and Dividing Rational Expressions
7.3 Adding and Subtracting Rational Expressions
7.4 Complex Fractions
7.5 Solving Rational Equations
7.6 Applications of Rational Equations
7.7 Variation
7.8 Review Exercises and Sample Exam

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Embark on an advanced journey with our Intermediate Algebra (IA) section, also known as Algebra 2. Delve into intricate topics, unravel complex equations, and enhance your mathematical prowess. Our meticulously crafted resources, combined with illustrative video tutorials, ensure a deeper understanding of algebraic concepts, bridging the gap between foundational knowledge and advanced applications. Whether you're gearing up for a challenging exam or simply eager to expand your algebraic horizons, our Intermediate Algebra section is tailored to meet your needs.

IA Chapter - Radical Expressions and Equations

1 Radicals
2 Simplifying Radical Expressions
3 Adding and Subtracting Radical Expressions
4 Multiplying and Dividing Radical Expressions
5 Rational Exponents
6 Solving Radical Equations
7 Complex Numbers
8 Review Exercises and Sample Exam

IA Chapter - Solving Quadratic Equations and Graphing Parabolas

1 Extracting Square Roots
2 Completing the Square
3 Quadratic Formula
4 Guidelines for Solving Quadratic Equations and Applications
5 Solving Equations Quadratic in Form
6 Graphing Parabolas
7 Complex Numbers and Complex Solutions
8 Review Exercises and Sample Exam

IA Chapter - Solving Inequalities
1 Absolute Value Equations
2 Absolute Value Inequalities
3 Quadratic Inequalities
4 Polynomial and Rational Inequalities

[ Intermediate Algebra Exam #1 Solutions ]
[ Intermediate Algebra Exam #2 Solutions ]
[ Intermediate Algebra Exam #3 Solutions ]
[ Intermediate Algebra Exam #4 Solutions ]

--- IA Sample Final Exam Ch. 1- 7 ---

IA Chapter - Graphing Basic Functions
1 Relations, Graphs, and Functions
2 Graphing the Basic Functions
3 Using Transformations to Graph Functions

IA Chapter - Exponential Equations and Logarithms

1 Function Composition
2 Inverse Functions
3 Exponential Functions and Their Graphs
4 The Natural Exponential Function
5 Logarithmic Functions
6 Graphing Logarithmic Functions
7 Change of Base Formula
8 Properties of the Logarithm
9 Solving Exponential Equations
10 Solving Logarithmic Equations
11 Applications
Compound Interest Problems
Exponential Growth/Decay

IA Chapter - Graphing Conic Sections

1. Distance and Midpoint Formulas
2. Circles
3. Ellipses
4. Hyperbolas
5. Solving Nonlinear Systems

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Factoring Lesson C by ChatGPT-4

Factoring Trinomials of the Form \(x^2 - bx - c\)

Factoring trinomials, which are polynomials with three terms, can initially seem challenging. However, with practice, it becomes routine. If a trinomial factors, it will factor into the product of two binomials.

Steps to Factor Trinomials:

Factor the first term: \(x^2 = x \cdot x\).
Factor the last term: Choose factors that multiply to obtain the last term and subtract to obtain the middle term.
Determine the signs: by adding or subtracting the product of the inner and outer terms.
Check by multiplying.

Example:

Factor the trinomial \(x^2 - 3x - 10\).

Factor the first term: \(x^2 = x \cdot x\).
Factor the last term: \(-10 = -5 \cdot 2\) and \(-5 + 2 = -3\), which is our middle term.
Determine the signs: One sign is positive and the other is negative because \(-5 + 2 = -3\).
Check by multiplying: \((x - 5)(x + 2) = x^2 - 3x - 10\).

So, \(x^2 - 3x - 10 = (x - 5)(x + 2)\).

Exercises:

Factor the following trinomials:

\(x^2 - 5x - 6\)
\(x^2 - 7x + 10\)
\(x^2 - 6x - 16\)
\(x^2 - 8x - 9\)
\(x^2 - 3x - 10\)
\(x^2 - 4x - 21\)
\(x^2 - 5x - 14\)
\(x^2 - 6x - 16\)
\(x^2 - 7x - 18\)
\(x^2 - 9x - 22\)

Solutions:

\(x^2 - 5x - 6 = (x - 6)(x + 1)\)
\(x^2 - 7x + 10 = (x - 2)(x - 5)\)
\(x^2 - 6x - 16 = (x - 8)(x + 2)\)
\(x^2 - 8x - 9 = (x - 9)(x + 1)\)
\(x^2 - 3x - 10 = (x - 5)(x + 2)\)
\(x^2 - 4x - 21 = (x - 7)(x + 3)\)
\(x^2 - 5x - 14 = (x - 7)(x + 2)\)
\(x^2 - 6x - 16 = (x - 8)(x + 2)\)
\(x^2 - 7x - 18 = (x - 9)(x + 2)\)
\(x^2 - 9x - 22 = (x - 11)(x + 2)\)

Remember to always check your results by multiplying the factors. If the product matches the original trinomial, then the factoring is correct.

Factoring Lesson B by ChatGPT-4

Factoring Trinomials of the Form \(x^2 - bx + c\)

Steps to Factor Trinomials:

Factor the first term: \(x^2 = x \cdot x\).
Factor the last term: Choose factors that multiply to obtain the last term and add to obtain the middle term.
Determine the signs: by adding or subtracting the product of the inner and outer terms.
Check by multiplying.

Example:

Factor the trinomial \(x^2 - 7x + 10\).

Factor the first term: \(x^2 = x \cdot x\).
Factor the last term: \(10 = 5 \cdot 2\) and \(5 + 2 = 7\), which is our middle term.
Determine the signs: Both signs are negative because \(-5 - 2 = -7\).
Check by multiplying: \((x - 5)(x - 2) = x^2 - 7x + 10\).

So, \(x^2 - 7x + 10 = (x - 5)(x - 2)\).

Exercises:

Factor the following trinomials:

\(x^2 - 5x + 6\)
\(x^2 - 9x + 20\)
\(x^2 - 7x + 10\)
\(x^2 - 11x + 30\)
\(x^2 - 8x + 15\)
\(x^2 - 10x + 21\)
\(x^2 - 6x + 8\)
\(x^2 - 12x + 35\)
\(x^2 - 15x + 56\)
\(x^2 - 13x + 40\)

Solutions:

\(x^2 - 5x + 6 = (x - 3)(x - 2)\)
\(x^2 - 9x + 20 = (x - 5)(x - 4)\)
\(x^2 - 7x + 10 = (x - 5)(x - 2)\)
\(x^2 - 11x + 30 = (x - 6)(x - 5)\)
\(x^2 - 8x + 15 = (x - 5)(x - 3)\)
\(x^2 - 10x + 21 = (x - 7)(x - 3)\)
\(x^2 - 6x + 8 = (x - 4)(x - 2)\)
\(x^2 - 12x + 35 = (x - 7)(x - 5)\)
\(x^2 - 15x + 56 = (x - 8)(x - 7)\)
\(x^2 - 13x + 40 = (x - 8)(x - 5)\)

Remember to always check your results by multiplying the factors. If the product matches the original trinomial, then the factoring is correct.

Factoring Lesson A by ChatGPT-4

Factoring Trinomials of the Form \(x^2 + bx + c\)

Steps to Factor Trinomials:

Factor the first term: \(x^2 = x \cdot x\).
Factor the last term: Choose factors that add or subtract to obtain the middle term.
Determine the signs: by adding or subtracting the product of the inner and outer terms.
Check by multiplying.

Example:

Factor the trinomial \(x^2 + 7x + 12\).

Factor the first term: \(x^2 = x \cdot x\).
Factor the last term: \(12 = 3 \cdot 4\) and \(3 + 4 = 7\), which is our middle term.
Determine the signs: Both signs are positive because \(+3 + 4 = +7\).
Check by multiplying: \((x + 3)(x + 4) = x^2 + 7x + 12\).

So, \(x^2 + 7x + 12 = (x + 3)(x + 4)\).

Exercises:

Factor the following trinomials:

\(x^2 + 5x + 6\)
\(x^2 + 9x + 20\)
\(x^2 + 7x + 10\)
\(x^2 + 11x + 30\)
\(x^2 + 8x + 15\)
\(x^2 + 10x + 21\)
\(x^2 + 6x + 8\)
\(x^2 + 12x + 35\)
\(x^2 + 15x + 56\)
\(x^2 + 13x + 40\)

Solutions:

\(x^2 + 5x + 6 = (x + 3)(x + 2)\)
\(x^2 + 9x + 20 = (x + 5)(x + 4)\)
\(x^2 + 7x + 10 = (x + 5)(x + 2)\)
\(x^2 + 11x + 30 = (x + 6)(x + 5)\)
\(x^2 + 8x + 15 = (x + 5)(x + 3)\)
\(x^2 + 10x + 21 = (x + 7)(x + 3)\)
\(x^2 + 6x + 8 = (x + 4)(x + 2)\)
\(x^2 + 12x + 35 = (x + 7)(x + 5)\)
\(x^2 + 15x + 56 = (x + 8)(x + 7)\)
\(x^2 + 13x + 40 = (x + 8)(x + 5)\)

Remember to always check your results by multiplying the factors. If the product matches the original trinomial, then the factoring is correct.

Comprehensive Algebra Video Guide: Step-by-Step Solutions & Tutorials

Click on a problem to see it worked out in YouTube.

Navigate the complexities of algebra with our extensive Algebra Video Compendium. From foundational concepts to advanced topics, our collection offers step-by-step video solutions on YouTube. Tailored to support the 'Intermediate Algebra' textbook by Flat World Knowledge, this visual guide simplifies algebraic learning, making it accessible and engaging for all.

1.1 Real Numbers and The Number Line