The Basic Processes of Algebra

Basic Definitions and Terminology

Algebra is a branch of mathematics where letters and symbols are used to represent numbers and quantities in equations and formulas.
An equation is a statement that two expressions are equal (e.g., 2x = 6). It expresses a fact or a principle.
A variable represents an unknown number or value. In algebra, variables are often represented by letters (e.g., x, y, z).
Constants are numbers that do not change their value.
Expressions are combinations of variables, numbers, and operations. They do not contain an equals (=) sign.
Coefficients are the numeric factors of variables (e.g., in 5x^2, 5 is the coefficient).

Algebraic Operations

Adding and subtracting like terms: When adding or subtracting algebraic expressions, you can only combine like terms. Like terms are terms that have the same variables raised to the same power.
Multiplication: Multiplication in algebra works under the same principle as in arithmetic. However, with algebra, you deal with variables in addition to numbers.
Division: Division is the operation that is the least commonly used with algebraic expressions. Division of algebraic expressions often involves the use of fractions.

Solving Simple Equations

Balancing the equation: To solve an equation means to find the value of the variable that makes the equation true. This is done by maintaining equality on both sides while simplifying the equation.
Use of inverse operations: An inverse operation undoes what the original operation does (addition-subtraction, multiplication-division).
Checking the solution: After finding a solution to an equation, always replace the variable in the original equation with the found value to confirm the solution.

Expanding and Factorising Algebraic Expressions

Expanding algebraic expressions: This refers to removing parentheses from equations by multiplying out.
Factorising algebraic expressions: Factorising is the reverse process of expanding. It involves expressing the expression as a product of factors.

Quadratic Equations

Quadratic equations have the form axe^2 + bx + c = 0. They are solved by factoring, completing the square, using the quadratic formula, or graphically.
Roots of a quadratic equation: The roots of a quadratic equation are the solutions to the equation. These are the values of x that make the equation true.

Inequality

Inequalities: Instead of an equals sign, inequalities have less than, greater than, less than or equal to, or greater than or equal to signs.
Solving inequalities: Inequalities are solved in almost the same way as equations, with one key difference. When multiplying or dividing both sides by a negative value, the inequality sign flips (e.g., if a > b, then -a < -b).

All the aforementioned points provide a foundation for Further Mathematics. Ensure you understand these basics to be successful in the more advanced topics.