Algebra: Algebraic Manipulation

This section covers the fundamental skills in algebra: manipulating expressions, using formulae, expanding brackets, and factorising. These skills are crucial for solving equations and simplifying mathematical statements.

1. Algebraic Expressions

An algebraic expression is a combination of numbers, variables (symbols representing unknown quantities), and mathematical operations (+, -, ×, ÷). Examples include: $3x + 5$, $2a - 7$, $\frac{4}{y} + 1$.

1.1 Simplifying Expressions

Simplifying an expression involves combining like terms. Like terms have the same variable part.

Example:

$3x + 2y - x + 5y = (3x - x) + (2y + 5y) = 2x + 7y$

2. Formulae

A formula is an equation that expresses a relationship between two or more quantities. We can manipulate formulae to solve for a specific variable.

Example: The area of a rectangle is given by $A = lw$. We can rearrange this to solve for $l$: $l = \frac{A}{w}$.

3. Expansion

Expansion involves removing the brackets from an expression using the distributive law. The distributive law states that $a(b + c) = ab + ac$.

3.1 Expanding Single Brackets

Example: Expand $2(x + 3)$

$2(x + 3) = 2 \times x + 2 \times 3 = 2x + 6$

3.2 Expanding Double Brackets

We use the distributive law four times to expand double brackets. The common terms are multiplied together.

Example: Expand $(x + 2)(x + 3)$

$= x(x + 3) + 2(x + 3)$

$= x \times x + x \times 3 + 2 \times x + 2 \times 3$

$= x^2 + 3x + 2x + 6$

$= x^2 + 5x + 6$

3.3 Expanding Using the FOIL Method

FOIL is a mnemonic for remembering the order of multiplication when expanding double brackets: First, Outer, Inner, Last.

Example: Expand $(2x - 1)(x + 4)$

First: $2x \times x = 2x^2$

Outer: $2x \times 4 = 8x$

Inner: $-1 \times x = -x$

Last: $-1 \times 4 = -4$

Combine: $2x^2 + 8x - x - 4 = 2x^2 + 7x - 4$

4. Factorisation

Factorisation involves writing an expression as a product of simpler expressions (factors). We look for common factors.

4.1 Factorising out a Common Factor

Example: Factorise $4x + 8y$

The common factor is $4$.

$4x + 8y = 4(x + 2y)$

4.2 Factorising Quadratic Expressions

For quadratic expressions of the form $ax^2 + bx + c$, we look for two numbers that multiply to $ac$ and add up to $b$.

Example: Factorise $x^2 + 5x + 6$

We need two numbers that multiply to $1 \times 6 = 6$ and add up to $5$. These numbers are $2$ and $3$.

$x^2 + 5x + 6 = (x + 2)(x + 3)$

4.3 Difference of Squares

The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$.

Example: Factorise $x^2 - 9$

$x^2 - 9 = x^2 - 3^2 = (x + 3)(x - 3)$

5. Summary Table

Topic	Description	Example
Algebraic Expressions	Combinations of variables and numbers.	$3x + 2y - x + 5y$
Formulae	Equations relating quantities, often rearranged.	$A = lw \implies l = \frac{A}{w}$
Expansion	Removing brackets using the distributive law.	$(x + 2)(x + 3) = x^2 + 5x + 6$
Factorisation	Expressing an expression as a product of factors.	$4x + 8y = 4(x + 2y)$