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  1. 1Basic Algebra
    1. 1.1Patterns
    2. 1.2Evaluating algebraic expressions
    3. 1.3Solving one - step equations: x + a = b
    4. 1.4Model and solve one-step linear equations: ax = b, x/a = b
    5. 1.5Solving two-step linear equations using addition and subtraction: ax + b = c
    6. 1.6Solving two-step linear equations using multiplication and division: x/a + b = c
    7. 1.7Solving two-step linear equations using distributive property: a(x + b) = c
    8. 1.8Solving literal equations
  2. 2Exponents
    1. 2.1Introduction to Exponents
    2. 2.2Using exponents to describe numbers
    3. 2.3Exponent rules
    4. 2.4Order of operations with exponents
    5. 2.5Using exponents to solve problems
    6. 2.6Product rule of exponents
    7. 2.7Quotient rule of exponents
    8. 2.8Power of a product rule
    9. 2.9Power of a quotient rule
    10. 2.10Power of a power rule
    11. 2.11Negative exponent rule
    12. 2.12Combining the exponent rules
    13. 2.13Scientific notation
    14. 2.14Solving for exponents
    15. 2.15Exponents: Product rule (a^x)(a^y) = a^(x+y)
    16. 2.16Exponents: Division rule (a^x / a^y) = a^(x-y)
    17. 2.17Exponents: Power rule (a^x)^y = a^(x * y)
    18. 2.18Exponents: Negative exponents
    19. 2.19Exponents: Zero exponent: a^0 = 1
    20. 2.20Exponents: Rational exponents
  3. 3Radicals
    1. 3.1Squares and square roots
    2. 3.2Estimating square roots
    3. 3.3Square and square roots
    4. 3.4Cubic and cube roots
    5. 3.5Evaluating and simplifying radicals
    6. 3.6Converting radicals to mixed radicals
    7. 3.7Converting radicals to entire radicals
    8. 3.8Adding and subtracting radicals
    9. 3.9Multiplying and dividing radicals
    10. 3.10Rationalize the denominator
    11. 3.11Convert between radicals and rational exponents
    12. 3.12Operations with radicals
    13. 3.13Conversion between entire radicals and mixed radicals
    14. 3.14Adding and subtracting radicals (Advanced)
    15. 3.15Multiplying radicals (Advanced)
  4. 4Unit Conversion
    1. 4.1Metric systems
    2. 4.2Imperial systems
    3. 4.3Conversions between metric and imperial systems
    4. 4.4Conversions involve squares and cubic
  5. 5Linear Equations
    1. 5.1Graphing linear relations
    2. 5.2Identifying proportional relationships
    3. 5.3Understanding graphs of linear relationships
    4. 5.4Understanding tables of values of linear relationships
    5. 5.5Applications of linear relationships
    6. 5.6Representing patterns in linear relations
    7. 5.7Reading linear relation graphs
    8. 5.8Solving linear equations by graphing
    9. 5.9Solving linear equations using multiplication and division
    10. 5.10Solving two-step linear equations: ax + b = c, x/a + b = c
    11. 5.11Solving linear equations using distributive property: a(x + b) = c
    12. 5.12Solving linear equations with variables on both sides
    13. 5.13Introduction to linear equations
    14. 5.14Introduction to nonlinear equations
    15. 5.15Special case of linear equations: Horizontal lines
    16. 5.16Special case of linear equations: Vertical lines
    17. 5.17Parallel line equation
    18. 5.18Perpendicular line equation
    19. 5.19Combination of both parallel and perpendicular line equations
    20. 5.20Applications of linear equations
  6. 6Linear Functions
    1. 6.1Distance formula: d=(x2x1)2+(y2y1)2d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
    2. 6.2Midpoint formula: M=(x1+x22,y1+y22)M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)
    3. 6.3Slope equation: m=y2y1x2x1m = \frac{y_2-y_1}{x_2- x_1}
    4. 6.4Slope intercept form: y = mx + b
    5. 6.5General form: Ax + By + C = 0
    6. 6.6Point-slope form: yy1=m(xx1)y - y_1 = m (x - x_1)
    7. 6.7Rate of change
    8. 6.8Graphing linear functions using table of values
    9. 6.9Graphing linear functions using x- and y-intercepts
    10. 6.10Graphing linear functions using various forms
    11. 6.11Graphing linear functions using a single point and slope
    12. 6.12Word problems of graphing linear functions
    13. 6.13Parallel and perpendicular lines in linear functions
    14. 6.14Applications of linear relations
  7. 7Introduction to Polynomials
    1. 7.1Characteristics of polynomials
    2. 7.2Equivalent expressions of polynomials
    3. 7.3What is a polynomial?
    4. 7.4Polynomial components
    5. 7.5Evaluating polynomials
    6. 7.6Using algebra tiles to factor polynomials
  8. 8Operations with Polynomials
    1. 8.1Adding and subtracting polynomials
    2. 8.2Multiplying and dividing monomials
    3. 8.3Multiplying polynomials by monomials
    4. 8.4Dividing polynomials by monomials
    5. 8.5Multiplying monomial by monomial
    6. 8.6Multiplying monomial by binomial
    7. 8.7Multiplying binomial by binomial
    8. 8.8Multiplying polynomial by polynomial
    9. 8.9Applications of polynomials
    10. 8.10Solving polynomial equations
    11. 8.11Word problems of polynomials
  9. 9Factoring Polynomials
    1. 9.1Common factors of polynomials
    2. 9.2Factoring polynomials by grouping
    3. 9.3Solving polynomials with unknown coefficients
    4. 9.4Solving polynomials with unknown constant terms
    5. 9.5Factoring polynomials: x^2 + bx + c
    6. 9.6Applications of polynomials: x^2 + bx + c
    7. 9.7Solving polynomials with the unknown "b" from ax2+bx+cax^2 + bx + c
    8. 9.8Factoring polynomials: ax2+bx+cax^2 + bx + c
    9. 9.9Factoring perfect square trinomials: (a + b)^2 = a^2 + 2ab + b^2 or (a - b)^2 = a^2 - 2ab + b^2
    10. 9.10Find the difference of squares: (a - b)(a + b) = (a^2 - b^2)
    11. 9.11Factor by taking out the greatest common factor
    12. 9.12Factor by grouping
    13. 9.13Factoring difference of squares: x^2 - y^2
    14. 9.14Factoring trinomials
    15. 9.15Factoring difference of cubes
    16. 9.16Factoring sum of cubes
  10. 10Polynomial Functions
    1. 10.1What is a polynomial function?
    2. 10.2Polynomial long division
    3. 10.3Polynomial synthetic division
    4. 10.4Remainder theorem
    5. 10.5Factor theorem
    6. 10.6Rational zero theorem
    7. 10.7Characteristics of polynomial graphs
    8. 10.8Multiplicities of polynomials
    9. 10.9Imaginary zeros of polynomials
    10. 10.10Determining the equation of a polynomial function
    11. 10.11Applications of polynomial functions
    12. 10.12Fundamental theorem of algebra
    13. 10.13Pascal's triangle
    14. 10.14Binomial theorem
    15. 10.15Descartes' rule of signs
  11. 11Quadratic Functions
    1. 11.1Introduction to quadratic functions
    2. 11.2Transformations of quadratic functions
    3. 11.3Quadratic function in general form: y = ax^2 + bx + c
    4. 11.4Quadratic function in vertex form: y = a(x-p)^2 + q
    5. 11.5Completing the square
    6. 11.6Converting from general to vertex form by completing the square
    7. 11.7Shortcut: Vertex formula
    8. 11.8Graphing parabolas for given quadratic functions
    9. 11.9Finding the quadratic functions for given parabolas
    10. 11.10Applications of quadratic functions
  12. 12Quadratic Equations
    1. 12.1Solving quadratic equations by factoring
    2. 12.2Solving quadratic equations by completing the square
    3. 12.3Solving quadratic equations using the quadratic formula
    4. 12.4Nature of roots of quadratic equations: The discriminant
    5. 12.5Solving polynomial equations by iteration
    6. 12.6Applications of quadratic equations
  13. 13System of Equations
    1. 13.1Determining number of solutions to linear equations
    2. 13.2Solving systems of linear equations by graphing
    3. 13.3Solving systems of linear equations by elimination
    4. 13.4Solving systems of linear equations by substitution
    5. 13.5Money related questions in linear equations
    6. 13.6Unknown number related questions in linear equations
    7. 13.7Distance and time related questions in linear equations
    8. 13.8Rectangular shape related questions in linear equations
    9. 13.9System of linear equations
    10. 13.10System of linear-quadratic equations
    11. 13.11System of quadratic-quadratic equations
    12. 13.12Solving 3 variable systems of equations by substitution
    13. 13.13Solving 3 variable systems of equations by elimination
    14. 13.14Solving 3 variable systems of equations with no or infinite solutions
    15. 13.15Word problems relating 3 variable systems of equations
  14. 14Inequalities
    1. 14.1Express linear inequalities graphically and algebraically
    2. 14.2Solving one-step linear inequalities
    3. 14.3Solving multi-step linear inequalities
    4. 14.4Compound inequalities
    5. 14.5Solving quadratic inequalities
    6. 14.6Inequalities of combined functions
    7. 14.7Solving polynomial inequalities
    8. 14.8Solving rational inequalities
    9. 14.9What is linear programming?
    10. 14.10Linear programming word problems
  15. 15Inequalities in Two Variables
    1. 15.1Graphing linear inequalities in two variables
    2. 15.2Graphing systems of linear inequalities
    3. 15.3Graphing quadratic inequalities in two variables
    4. 15.4Graphing systems of quadratic inequalities
    5. 15.5Applications of inequalities
  16. 16Absolute Value
    1. 16.1Introduction to absolute value
    2. 16.2Absolute value functions
    3. 16.3Solving absolute value equations
    4. 16.4Solving absolute value inequalities
  17. 17Set Theory
    1. 17.1Set notation
    2. 17.2Set builder notation
    3. 17.3Intersection and union of 2 sets
    4. 17.4Intersection and union of 3 sets
    5. 17.5Interval notations
  18. 18Relations and Functions
    1. 18.1Relationship between two variables
    2. 18.2Understand relations between x- and y-intercepts
    3. 18.3Domain and range of a function
    4. 18.4Identifying functions
    5. 18.5Function notation
    6. 18.6Function notation (Advanced)
    7. 18.7Operations with functions
    8. 18.8Adding functions
    9. 18.9Subtracting functions
    10. 18.10Multiplying functions
    11. 18.11Dividing functions
    12. 18.12Composite functions
    13. 18.13Inverse functions
    14. 18.14One to one functions
    15. 18.15Difference quotient: applications of functions
    16. 18.16Transformations of functions: Horizontal translations
    17. 18.17Transformations of functions: Vertical translations
    18. 18.18Reflection across the y-axis: y = f(-x)
    19. 18.19Reflection across the x-axis: y = -f(x)
    20. 18.20Transformations of functions: Horizontal stretches
    21. 18.21Transformations of functions: Vertical stretches
    22. 18.22Combining transformations of functions
    23. 18.23Even and odd functions
    24. 18.24Direct variation
    25. 18.25Inverse variation
    26. 18.26Joint and combined variation
  19. 19Rational Equations and Expressions
    1. 19.1Simplifying rational expressions and restrictions
    2. 19.2Adding and subtracting rational expressions
    3. 19.3Multiplying rational expressions
    4. 19.4Dividing rational expressions
    5. 19.5Solving rational equations
    6. 19.6Applications of rational equations
    7. 19.7Simplifying complex fractions
    8. 19.8Partial fraction decomposition
  20. 20Rational Functions
    1. 20.1What is a rational function?
    2. 20.2Point of discontinuity
    3. 20.3Vertical asymptote
    4. 20.4Horizontal asymptote
    5. 20.5Slant asymptote
    6. 20.6Graphs of rational functions
  21. 21Radical Functions and Equations
    1. 21.1Basic radical functions
    2. 21.2Transformations of radical functions
    3. 21.3Square root of a function
    4. 21.4Solving radical equations
  22. 22Reciprocal Functions
    1. 22.1Graphing reciprocals of linear functions
    2. 22.2Graphing reciprocals of quadratic functions
  23. 23Piecewise Functions
    1. 23.1Evaluating piecewise functions
    2. 23.2Graphing piecewise linear functions
    3. 23.3Graphing piecewise non-linear functions
  24. 24Exponential Functions
    1. 24.1Solving exponential equations using exponent rules
    2. 24.2Graphing exponential functions
    3. 24.3Graphing transformations of exponential functions
    4. 24.4Finding an exponential function given its graph
    5. 24.5Exponential growth and decay by a factor
    6. 24.6Exponential decay: Half-life
    7. 24.7Exponential growth and decay by percentage
    8. 24.8Finance: Compound interest
    9. 24.9Continuous growth and decay
    10. 24.10Finance: Future value and present value
  25. 25Logarithm
    1. 25.1What is a logarithm?
    2. 25.2Converting from logarithmic form to exponential form
    3. 25.3Evaluating logarithms without a calculator
    4. 25.4Common logarithms
    5. 25.5Natural log: ln
    6. 25.6Evaluating logarithms using change-of-base formula
    7. 25.7Converting from exponential form to logarithmic form
    8. 25.8Solving exponential equations with logarithms
    9. 25.9Product rule of logarithms
    10. 25.10Quotient rule of logarithms
    11. 25.11Combining product rule and quotient rule in logarithms
    12. 25.12Evaluating logarithms using logarithm rules
    13. 25.13Solving logarithmic equations
    14. 25.14Graphing logarithmic functions
    15. 25.15Finding a logarithmic function given its graph
    16. 25.16Logarithmic scale: Richter scale (earthquake)
    17. 25.17Logarithmic scale: pH scale
    18. 25.18Logarithmic scale: dB scale
  26. 26Sequences and Series
    1. 26.1Arithmetic sequences
    2. 26.2Arithmetic series
    3. 26.3Geometric sequences
    4. 26.4Geometric series
    5. 26.5Infinite geometric series
    6. 26.6Sigma notation
    7. 26.7Arithmetic mean vs. Geometric mean
    8. 26.8Linear sequences
    9. 26.9Quadratic sequences
  27. 27Conic Sections
    1. 27.1Conics - Parabola
    2. 27.2Conics - Ellipse
    3. 27.3Conics - Circle
    4. 27.4Conics - Hyperbola
  28. 28Complex Numbers
    1. 28.1Introduction to imaginary numbers
    2. 28.2Complex numbers and complex planes
    3. 28.3Adding and subtracting complex numbers
    4. 28.4Complex conjugates
    5. 28.5Multiplying and dividing complex numbers
    6. 28.6Distance and midpoint of complex numbers
    7. 28.7Angle and absolute value of complex numbers
    8. 28.8Polar form of complex numbers
    9. 28.9Operations on complex numbers in polar form
  29. 29Vectors
    1. 29.1Introduction to vectors
    2. 29.2Magnitude of a vector
    3. 29.3Direction angle of a vector
    4. 29.4Scalar multiplication
    5. 29.5Equivalent vectors
    6. 29.6Adding and subtracting vectors in component form
    7. 29.7Operations on vectors in magnitude and direction form
    8. 29.8Unit Vector
    9. 29.9Word problems on vectors
  30. 30Matrices
    1. 30.1Notation of matrices
    2. 30.2Adding and subtracting matrices
    3. 30.3Scalar multiplication
    4. 30.4Matrix multiplication
    5. 30.5The three types of matrix row operations
    6. 30.6Representing a linear system as a matrix
    7. 30.7Solving a linear system with matrices using Gaussian elimination
    8. 30.8Zero matrix
    9. 30.9Identity matrix
    10. 30.10Properties of matrix addition
    11. 30.11Properties of scalar multiplication
    12. 30.12Properties of matrix multiplication
    13. 30.13The determinant of a 2 x 2 matrix
    14. 30.14The determinant of a 3 x 3 matrix (General & Shortcut Method)
    15. 30.15The inverse of a 2 x 2 matrix
    16. 30.16The inverse of 3 x 3 matrices with matrix row operations
    17. 30.17The inverse of 3 x 3 matrix with determinants and adjugate
    18. 30.182 x 2 invertible matrix
    19. 30.19Solving linear systems using Cramer's Rule
    20. 30.20Solving linear systems using 2 x 2 inverse matrices
    21. 30.21Transforming vectors with matrices
    22. 30.22Transforming shapes with matrices
    23. 30.23Finding the transformation matrix

What is Algebra?

Algebra is the study of mathematical symbols and the rules surrounding the manipulation of these symbols. The symbols are used to represent quantities and numbers within equations and formulae.

We use symbols in algebra because of the unknown values, known as variables. If there is no known value (number) like 10 or 5, we place a letter or symbol (usually X or a Y ) to represent that the value is unknown. In contrast to this, there are elements of algebraic equations called "constants". These get their name because the value remains the same throughout. For example, the number 23 will always represent 23 units and never another value.

As an Example: X + 23 = 28

In the basic equation above, the variable (x) is the number required to make this sum work. All we need to do is to add another number to 23 (constant) so that we can get 28 (constant). Adding 5 to 23, gives you 28, so now we know that the value of x is 5.

Algebra is considered a unifying concept of almost all elements of math and as such, it stretches far beyond elementary equations and into the study of things like, radicals, integers, linear systems, exponent rules, quadratic equations, the Pythagorean Theorem and much more.

Who Invented Algebra?

The roots of algebra can be traced back to the ancient babylonian era. The babylonians had advanced systems of math that allowed them to calculate things in an algorithmic fashion. For the longest time, It was thought that the "father of algebra" was Diophantus, an alexandrian greek mathematician. This has recently been challenged though, with historians and mathematicians suggesting that the title should be with al-Khwarizmi, the founder of the discipline of al-jabr. It's worth noting that the term "al-jabr", meaning reunion of broken parts, is where we get the word algebra from.

How to Learn Algebra?

With algebra, like anything in math, the best way to learn is with repetition. Practice makes perfect, so make time outside of school/college hours to review your class notes. If you aren't taking notes in class, you really should be. These notes will not only help you retain the information but they can also be used to help you identify any areas of weakness that you have in relation to algebra. It's important that you know where your strengths and weaknesses are so that you can build effective revision strategies ahead of your upcoming exams. Highlight key areas for improvement and get yourself into a good study habit.

When it comes time to revise, use StudyPug's vast collection of algebra lessons to help you. With step-by-step examples for even the toughest algebra problems, we offer easy practice solutions for high school algebra content through to more complex college level algebra. With your very own algebra tutor, you can select the lessons that are relevant to you, making for a tailored study session that works to strengthen the weaker areas of your knowledge.

If you're looking for algebra homework help, we've got you covered. Additionally, we'll also assist you with your algebra exam prep, covering all the content you'd expect to find on the actual papers. As mentioned above, our algebra tutoring sessions cover all skill levels, so if you're looking for an introduction to algebra, we can get you started. Our content will walk you through each aspect of algebra and when you're ready, you can progress to the next step. In no time at all, you'll be solving equations, understanding algebra formulas and so much more! This is because our content is delivered via video tutorials that can be paused, rewound, and fast-forwarded, ensuring that you can learn at your own pace and will never get left behind.

Many of our students find the video format is a lot more palatable and easier to follow than traditional textbooks. That's because our online content avoids overly complex terminology and presents it in a way that's much easier to understand. The videos are presented by knowledgeable math teachers who have worked to ensure that their video content covers the same content found in class and on exams.

What Grade Level is Algebra?

Algebra is one of the foundations of mathematics and as such, it's taught in one form or another at all grade levels. It is taught to young students once they have learnt how to find the pattern in numbers, and with each passing grade, the students continue to build their understanding of the more complex elements eventually learning abstract algebra at college and university.

Below we have listed the common algebra focused courses and at what grade you can expect to study them

  • Pre-algebra: Usually Studied in 5th or 6th Grade
  • Basic Algebra: Usually Studied in 7th or 8th Grade
  • Algebra I: Usually Studied in 9th Grade
  • Algebra II: Usually Studied in 11th Grade
  • Intermediate Algebra: Taken Before Any University Level Math, E.G. College Algebra or Calculus
  • College Algebra: First Year of College or University

How Can I Solve and Simplify Algebraic Equations Without a Calculator?

In order to solve algebraic equations without the use of a calculator, you're going to need to understand algebra instead of simply memorizing the formulae. Once you have a firm grasp on how the math works, you'll be able to tackle complex math problems without a calculator.

You should consider browsing our website for helpful videos that break down math into step-by-step examples so that you not only memorize them but actually understand them too. We also have useful tips on how to solve other problems with calculators, such as estimating square roots, percent of a number, and evaluating logarithms.

It's also worth noting that math hasn't changed and there was a time before calculators. Track down some older textbooks and you may find some useful tips for solving math problems with only a pen, some paper, and a little mental arithmetic.

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