[Leejohnston]

Algebra Essentials — Expand, Factor, Solve (With Examples)

Algebra is the language of mathematics. 
Once you master the basic tools — expanding expressions, factorising, and solving equations — every other branch of maths becomes far easier.

This thread gives a clean introduction to the essential ideas, with examples and practice questions.

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1. What Is Algebra?

Algebra uses:
• letters (variables) 
• numbers 
• symbols 

…to create rules and relationships.

Example: 
A = πr² describes the area of a circle for ANY radius.

Algebra lets us:
• generalise 
• solve unknowns 
• spot patterns 
• build models 
• describe real systems 

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2. Expanding Brackets

Use the distributive law:

a(b + c) = ab + ac

Examples:

Code:

2(x + 5) = 2x + 10
3(2y - 7) = 6y - 21

Double brackets (FOIL):

(a + b)(c + d)

Example:

Code:

(x + 3)(x + 2)
= x² + 2x + 3x + 6
= x² + 5x + 6

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3. Factorising (Opposite of Expanding)

Factorising takes an expression and pulls out common factors.

a(b + c) is the factorised form of ab + ac.

Examples:

Common factor:

Code:

6x + 12 = 6(x + 2)
5y - 20 = 5(y - 4)

Quadratic factorising:

Code:

x² + 5x + 6 = (x + 2)(x + 3)
x² - 3x - 10 = (x - 5)(x + 2)

Tips:
• Look for two numbers that multiply to the constant term 
• …and add to the x-coefficient 

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4. Solving Linear Equations

Goal: get the variable by itself.

Example 1:

Code:

3x + 5 = 14
3x = 9
x = 3

Example 2:

Code:

7 - 2y = 1
-2y = -6
y = 3

Always:
• move terms across using opposite operations 
• keep the equation balanced 

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5. Solving Quadratic Equations

Method 1: Factorising

Code:

x² + 5x + 6 = 0
(x + 2)(x + 3) = 0
x = -2 or x = -3

Method 2: Using the quadratic formula

Code:

x = (-b ± √(b² - 4ac)) / (2a)

Example:

Code:

x² - 4x + 1 = 0
a=1, b=-4, c=1

Substitute and solve.

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6. Number Theory Basics

Prime numbers: 
Numbers divisible only by 1 and themselves. 
Examples: 2, 3, 5, 7, 11, 13…

Factors and multiples: 
12 → factors: 1, 2, 3, 4, 6, 12 
Multiple of 4 → 4, 8, 12, 16…

Highest Common Factor (HCF): 
Largest number that divides both.

Lowest Common Multiple (LCM): 
Smallest number in both multiples lists.

Prime factorisation:

Example: 
24 = 2 × 2 × 2 × 3 = 2³ × 3

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7. Indices (Powers) — Quick Rules

xᵃ × xᵇ = xᵃ⁺ᵇ 
xᵃ ÷ xᵇ = xᵃ⁻ᵇ 
(xᵃ)ᵇ = xᵃᵇ 
x⁰ = 1 
x⁻ⁿ = 1/xⁿ

Examples:

Code:

2³ × 2² = 2⁵ = 32
y⁵ ÷ y² = y³
(3²)³ = 3⁶

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8. Key Algebra Mistakes to Avoid

❌ Expanding incorrectly 
✔ 3(x + 4) = 3x + 12 (not 3x + 4)

❌ Mixing up factorising and solving 
✔ factorising rewrites the expression 
✔ solving finds x

❌ Forgetting negative signs 
✔ brackets help: -(x - 5) = -x + 5

❌ Thinking √(a + b) = √a + √b 
✔ It does NOT

❌ Using the wrong two numbers in quadratics 
✔ Check: multiply to constant, add to x-term

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9. Practice Questions

Expanding 
1. 4(x - 3) 
2. (x + 5)(x - 1)

Factorising 
3. 8y + 12 
4. x² + 7x + 12

Solving 
5. 5x - 9 = 16 
6. 3y + 4 = 2y - 6

Number Theory 
7. Prime factors of 60 
8. HCF of 24 and 36

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Summary

This thread introduced:
• expanding brackets 
• factorising 
• solving equations 
• quadratic methods 
• prime numbers & factors 
• index rules 
• practice exercises 

Algebra is the foundation of nearly all mathematics — master these tools and everything else becomes easier.