Algebra Basics

Introduction to Algebra

• Algebra is the part of mathematics in which variables and constants are related using numerical operations.
• An algebraic expression is a mathematical phrase that can contain numbers, operators, and at least one variable. An example would be 2n + 3.
• Variables are letters used to represent unspecified numbers or values.
• Constants are items that do not change. They include numbers or set values.

Understanding Equations

• Equations are mathematical statements that assert the equality of two expressions. Examples include 2x + 5 = 0.
• Equating the two sides helps solve the equation to find the value of the variable.
• Algebra requires the adaptation of the BODMAS rule. It stands for Brackets, Orders or powers, Division and Multiplication, and Addition and Subtraction.

Algebraic Operations

• Adding or subtracting like terms is important in simplifying algebraic expressions.
• When multiplying terms, multiply the coefficient (numerical part) and add the exponents of the variables, if they are similar.
• When dividing terms, divide the coefficient, and subtract the exponents of the variables, if they are similar.
• In the case of exponentiation, when the base numbers are the same, add the exponents together.

Solving Equations

• Balancing the equation is central to solving equations. This means if you perform an operation (addition, subtraction, etc.) on one side, you must do it to the other side as well.
• One can practise memory of algebra formulae with the Quadratic Formula which is used to solve quadratic equations: x = [ -b ± sqrt(b^2 - 4ac) ] / 2a

Factorising and Expanding

• Factorising involves writing the given terms as a product of their factors.
• Expanding is the process of removing the brackets from an algebraic expression.

Linear and Quadratic Equations

• Linear equations in a single variable are ones where the variable has an exponent of "1", which is typically not shown (e.g., x + 2 = 0).
• Quadratic equations appear in the format of ax^2 + bx + c = 0, a ≠ 0. Solutions can be real or complex which can be found using a variety of methods such as factorisation, completing the square, or using the quadratic formula.

Remember, practising these fundamentals regularly will sharpen your algebraic skills and boost your problem-solving ability.