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Master equations, functions, and algebraic applications

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Algebra Topics

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Basic Algebra
Patterns
Evaluating algebraic expressions
Solving one - step equations: x + a = b
Model and solve one-step linear equations: ax = b, x/a = b
Solving two-step linear equations using addition and subtraction: ax + b = c
Solving two-step linear equations using multiplication and division: x/a + b = c
Solving two-step linear equations using distributive property: a(x + b) = c
Solving literal equations
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Exponents
Introduction to Exponents
Using exponents to describe numbers
Exponent rules
Order of operations with exponents
Using exponents to solve problems
Product rule of exponents
Quotient rule of exponents
Power of a product rule
Power of a quotient rule
Power of a power rule
Negative exponent rule
Combining the exponent rules
Scientific notation
Solving for exponents
Exponents: Product rule (a^x)(a^y) = a^(x+y)
Exponents: Division rule (a^x / a^y) = a^(x-y)
Exponents: Power rule (a^x)^y = a^(x * y)
Exponents: Negative exponents
Exponents: Zero exponent: a^0 = 1
Exponents: Rational exponents
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Radicals
Squares and square roots
Estimating square roots
Square and square roots
Cubic and cube roots
Evaluating and simplifying radicals
Converting radicals to mixed radicals
Converting radicals to entire radicals
Adding and subtracting radicals
Multiplying and dividing radicals
Rationalize the denominator
Convert between radicals and rational exponents
Operations with radicals
Conversion between entire radicals and mixed radicals
Adding and subtracting radicals (Advanced)
Multiplying radicals (Advanced)
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Unit Conversion
Metric systems
Imperial systems
Conversions between metric and imperial systems
Conversions involve squares and cubic
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Linear Equations
Graphing linear relations
Identifying proportional relationships
Understanding graphs of linear relationships
Understanding tables of values of linear relationships
Applications of linear relationships
Representing patterns in linear relations
Reading linear relation graphs
Solving linear equations by graphing
Solving linear equations using multiplication and division
Solving two-step linear equations: ax + b = c, x/a + b = c
Solving linear equations using distributive property: a(x + b) = c
Solving linear equations with variables on both sides
Introduction to linear equations
Introduction to nonlinear equations
Special case of linear equations: Horizontal lines
Special case of linear equations: Vertical lines
Parallel line equation
Perpendicular line equation
Combination of both parallel and perpendicular line equations
Applications of linear equations
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Linear Functions
Distance formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}d=(x2​−x1​)2+(y2​−y1​)2​
Midpoint formula: M=(x1+x22,y1+y22)M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)M=(2x1​+x2​​,2y1​+y2​​)
Slope equation: m=y2−y1x2−x1m = \frac{y_2-y_1}{x_2- x_1}m=x2​−x1​y2​−y1​​
Slope intercept form: y = mx + b
General form: Ax + By + C = 0
Point-slope form: y−y1=m(x−x1)y - y_1 = m (x - x_1)y−y1​=m(x−x1​)
Rate of change
Graphing linear functions using table of values
Graphing linear functions using x- and y-intercepts
Graphing linear functions using various forms
Graphing linear functions using a single point and slope
Word problems of graphing linear functions
Parallel and perpendicular lines in linear functions
Applications of linear relations
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Introduction to Polynomials
Characteristics of polynomials
Equivalent expressions of polynomials
What is a polynomial?
Polynomial components
Evaluating polynomials
Using algebra tiles to factor polynomials
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Operations with Polynomials
Adding and subtracting polynomials
Multiplying and dividing monomials
Multiplying polynomials by monomials
Dividing polynomials by monomials
Multiplying monomial by monomial
Multiplying monomial by binomial
Multiplying binomial by binomial
Multiplying polynomial by polynomial
Applications of polynomials
Solving polynomial equations
Word problems of polynomials
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Factoring Polynomials
Common factors of polynomials
Factoring polynomials by grouping
Solving polynomials with unknown coefficients
Solving polynomials with unknown constant terms
Factoring polynomials: x^2 + bx + c
Applications of polynomials: x^2 + bx + c
Solving polynomials with the unknown "b" from ax2+bx+cax^2 + bx + cax2+bx+c
Factoring polynomials: ax2+bx+cax^2 + bx + cax2+bx+c
Factoring perfect square trinomials: (a + b)^2 = a^2 + 2ab + b^2 or (a - b)^2 = a^2 - 2ab + b^2
Find the difference of squares: (a - b)(a + b) = (a^2 - b^2)
Factor by taking out the greatest common factor
Factor by grouping
Factoring difference of squares: x^2 - y^2
Factoring trinomials
Factoring difference of cubes
Factoring sum of cubes
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Polynomial Functions
What is a polynomial function?
Polynomial long division
Polynomial synthetic division
Remainder theorem
Factor theorem
Rational zero theorem
Characteristics of polynomial graphs
Multiplicities of polynomials
Imaginary zeros of polynomials
Determining the equation of a polynomial function
Applications of polynomial functions
Fundamental theorem of algebra
Pascal's triangle
Binomial theorem
Descartes' rule of signs
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Quadratic Functions
Introduction to quadratic functions
Transformations of quadratic functions
Quadratic function in general form: y = ax^2 + bx + c
Quadratic function in vertex form: y = a(x-p)^2 + q
Completing the square
Converting from general to vertex form by completing the square
Shortcut: Vertex formula
Graphing parabolas for given quadratic functions
Finding the quadratic functions for given parabolas
Applications of quadratic functions
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Quadratic Equations
Solving quadratic equations by factoring
Solving quadratic equations by completing the square
Solving quadratic equations using the quadratic formula
Nature of roots of quadratic equations: The discriminant
Solving polynomial equations by iteration
Applications of quadratic equations
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System of Equations
Determining number of solutions to linear equations
Solving systems of linear equations by graphing
Solving systems of linear equations by elimination
Solving systems of linear equations by substitution
Money related questions in linear equations
Unknown number related questions in linear equations
Distance and time related questions in linear equations
Rectangular shape related questions in linear equations
System of linear equations
System of linear-quadratic equations
System of quadratic-quadratic equations
Solving 3 variable systems of equations by substitution
Solving 3 variable systems of equations by elimination
Solving 3 variable systems of equations with no or infinite solutions
Word problems relating 3 variable systems of equations
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Inequalities
Express linear inequalities graphically and algebraically
Solving one-step linear inequalities
Solving multi-step linear inequalities
Compound inequalities
Solving quadratic inequalities
Inequalities of combined functions
Solving polynomial inequalities
Solving rational inequalities
What is linear programming?
Linear programming word problems
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Inequalities in Two Variables
Graphing linear inequalities in two variables
Graphing systems of linear inequalities
Graphing quadratic inequalities in two variables
Graphing systems of quadratic inequalities
Applications of inequalities
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Absolute Value
Introduction to absolute value
Absolute value functions
Solving absolute value equations
Solving absolute value inequalities
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Set Theory
Set notation
Set builder notation
Intersection and union of 2 sets
Intersection and union of 3 sets
Interval notations
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Relations and Functions
Relationship between two variables
Understand relations between x- and y-intercepts
Domain and range of a function
Identifying functions
Function notation
Function notation (Advanced)
Operations with functions
Adding functions
Subtracting functions
Multiplying functions
Dividing functions
Composite functions
Inverse functions
One to one functions
Difference quotient: applications of functions
Transformations of functions: Horizontal translations
Transformations of functions: Vertical translations
Reflection across the y-axis: y = f(-x)
Reflection across the x-axis: y = -f(x)
Transformations of functions: Horizontal stretches
Transformations of functions: Vertical stretches
Combining transformations of functions
Even and odd functions
Direct variation
Inverse variation
Joint and combined variation
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Rational Equations and Expressions
Simplifying rational expressions and restrictions
Adding and subtracting rational expressions
Multiplying rational expressions
Dividing rational expressions
Solving rational equations
Applications of rational equations
Simplifying complex fractions
Partial fraction decomposition
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Rational Functions
What is a rational function?
Point of discontinuity
Vertical asymptote
Horizontal asymptote
Slant asymptote
Graphs of rational functions
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Radical Functions and Equations
Basic radical functions
Transformations of radical functions
Square root of a function
Solving radical equations
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Reciprocal Functions
Graphing reciprocals of linear functions
Graphing reciprocals of quadratic functions
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Piecewise Functions
Evaluating piecewise functions
Graphing piecewise linear functions
Graphing piecewise non-linear functions
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Exponential Functions
Solving exponential equations using exponent rules
Graphing exponential functions
Graphing transformations of exponential functions
Finding an exponential function given its graph
Exponential growth and decay by a factor
Exponential decay: Half-life
Exponential growth and decay by percentage
Finance: Compound interest
Continuous growth and decay
Finance: Future value and present value
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Logarithm
What is a logarithm?
Converting from logarithmic form to exponential form
Evaluating logarithms without a calculator
Common logarithms
Natural log: ln
Evaluating logarithms using change-of-base formula
Converting from exponential form to logarithmic form
Solving exponential equations with logarithms
Product rule of logarithms
Quotient rule of logarithms
Combining product rule and quotient rule in logarithms
Evaluating logarithms using logarithm rules
Solving logarithmic equations
Graphing logarithmic functions
Finding a logarithmic function given its graph
Logarithmic scale: Richter scale (earthquake)
Logarithmic scale: pH scale
Logarithmic scale: dB scale
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Sequences and Series
Arithmetic sequences
Arithmetic series
Geometric sequences
Geometric series
Infinite geometric series
Sigma notation
Arithmetic mean vs. Geometric mean
Linear sequences
Quadratic sequences
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Conic Sections
Conics - Parabola
Conics - Ellipse
Conics - Circle
Conics - Hyperbola
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Complex Numbers
Introduction to imaginary numbers
Complex numbers and complex planes
Adding and subtracting complex numbers
Complex conjugates
Multiplying and dividing complex numbers
Distance and midpoint of complex numbers
Angle and absolute value of complex numbers
Polar form of complex numbers
Operations on complex numbers in polar form
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Vectors
Introduction to vectors
Magnitude of a vector
Direction angle of a vector
Scalar multiplication
Equivalent vectors
Adding and subtracting vectors in component form
Operations on vectors in magnitude and direction form
Unit Vector
Word problems on vectors
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Matrices
Notation of matrices
Adding and subtracting matrices
Scalar multiplication
Matrix multiplication
The three types of matrix row operations
Representing a linear system as a matrix
Solving a linear system with matrices using Gaussian elimination
Zero matrix
Identity matrix
Properties of matrix addition
Properties of scalar multiplication
Properties of matrix multiplication
The determinant of a 2 x 2 matrix
The determinant of a 3 x 3 matrix (General & Shortcut Method)
The inverse of a 2 x 2 matrix
The inverse of 3 x 3 matrices with matrix row operations
The inverse of 3 x 3 matrix with determinants and adjugate
2 x 2 invertible matrix
Solving linear systems using Cramer's Rule
Solving linear systems using 2 x 2 inverse matrices
Transforming vectors with matrices
Transforming shapes with matrices
Finding the transformation matrix
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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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