IGCSE Maths Algebra Explained: From Basics to Advanced

Algebra is fundamental to IGCSE Mathematics. This progressive guide takes you from basic algebraic concepts to advanced topics, explaining expressions, equations, factorizing, and functions step-by-step with clear examples.

Basics: What is Algebra?

Algebra uses letters (variables) to represent numbers. This allows us to:

• Write general rules
• Solve problems with unknown values
• Work with formulas
• Express relationships

Key Terms:

• Variable: Letter representing unknown (x, y, a, b)
• Constant: Fixed number (5, -3, ½)
• Coefficient: Number multiplying variable (in 3x, 3 is coefficient)
• Term: Single part of expression (3x, 5, -2y)
• Expression: Combination of terms (3x + 5 - 2y)

Basic Operations

Adding and Subtracting

• Combine like terms (same variable)
• 3x + 5x = 8x
• 2a - a = a
• Can’t combine different variables: 3x + 2y stays as is

Multiplying

• Multiply coefficients, add powers
• 3x × 2x = 6x²
• 4a × 5a² = 20a³
• x × x = x²

Dividing

• Divide coefficients, subtract powers
• 6x² ÷ 2x = 3x
• 12a³ ÷ 4a = 3a²
• x⁵ ÷ x² = x³

Expanding Brackets

Single Brackets

• Multiply each term inside by term outside
• 3(x + 2) = 3x + 6
• -2(3x - 5) = -6x + 10
• x(2x + 3) = 2x² + 3x

Double Brackets (FOIL)

• First, Outer, Inner, Last
• (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6
• (x - 1)(x + 4) = x² + 4x - x - 4 = x² + 3x - 4

Examples:

• (2x + 1)(x - 3) = 2x² - 6x + x - 3 = 2x² - 5x - 3
• (x + 5)² = (x + 5)(x + 5) = x² + 10x + 25

Factorizing

Common Factor

• Find what’s common to all terms
• 6x + 9 = 3(2x + 3)
• 4x² - 8x = 4x(x - 2)
• 12a + 18b = 6(2a + 3b)

Quadratic Factorizing

• Find two numbers that multiply to constant and add to coefficient
• x² + 5x + 6 = (x + 2)(x + 3) [2×3=6, 2+3=5]
• x² - 7x + 12 = (x - 3)(x - 4) [-3×-4=12, -3+-4=-7]
• x² - 9 = (x + 3)(x - 3) [difference of two squares]

Difference of Two Squares

• a² - b² = (a + b)(a - b)
• x² - 16 = (x + 4)(x - 4)
• 9x² - 25 = (3x + 5)(3x - 5)

Solving Equations

Linear Equations

• Get variable on one side, numbers on other
• Do same to both sides
• Example: 3x + 5 = 14
  • Subtract 5: 3x = 9
  • Divide by 3: x = 3

Equations with Brackets

• Expand brackets first
• Then solve as normal
• Example: 2(x + 3) = 10
  • Expand: 2x + 6 = 10
  • Solve: 2x = 4, x = 2

Equations with Fractions

• Clear fractions by multiplying by denominator
• Then solve as normal
• Example: (x + 1)/3 = 4
  • Multiply by 3: x + 1 = 12
  • Solve: x = 11

Quadratic Equations

Solving by Factorizing

• Factorize quadratic
• Set each bracket = 0
• Solve each equation
• Example: x² - 5x + 6 = 0
  • Factorize: (x - 2)(x - 3) = 0
  • Solutions: x = 2 or x = 3

Quadratic Formula

• Use when factorizing difficult
• x = (-b ± √(b² - 4ac)) / 2a
• For ax² + bx + c = 0
• Example: 2x² + 5x - 3 = 0
  • a=2, b=5, c=-3
  • x = (-5 ± √(25 + 24)) / 4
  • x = (-5 ± 7) / 4
  • x = 0.5 or x = -3

Simultaneous Equations

Substitution Method

• Rearrange one equation
• Substitute into other equation
• Solve for one variable
• Substitute back to find other

Example:

• x + y = 5 and 2x - y = 1
• From first: y = 5 - x
• Substitute: 2x - (5 - x) = 1
• Solve: 2x - 5 + x = 1, so 3x = 6, x = 2
• Substitute: y = 5 - 2 = 3

Elimination Method

• Add or subtract equations to eliminate variable
• Solve for remaining variable
• Substitute to find other

Functions

What is a Function?

• Rule that maps input to output
• f(x) = … (function of x)
• Each input has one output
• Domain: possible inputs
• Range: possible outputs

Function Notation:

• f(x) = 2x + 3
• f(2) means substitute x = 2
• f(2) = 2(2) + 3 = 7

Composite Functions:

• f(g(x)): apply g first, then f
• Example: f(x) = x + 1, g(x) = 2x
• f(g(3)) = f(6) = 7

Inverse Functions:

• f⁻¹(x): reverses function
• If f(x) = 2x + 3, then f⁻¹(x) = (x - 3)/2
• Check: f(f⁻¹(x)) = x

Advanced Topics

Inequalities

• Similar to equations but with <, >, ≤, ≥
• Solve same way but flip sign if multiply/divide by negative
• Example: 3x + 2 > 11
  • 3x > 9
  • x > 3

Sequences

• Arithmetic: add same number each time
• Geometric: multiply by same number each time
• nth term formulas
• Example: 2, 5, 8, 11… (arithmetic, +3)

Common Mistakes to Avoid

1. Sign Errors

• Be careful with negative signs
• Check when expanding brackets
• Verify when solving equations

2. Not Combining Like Terms

• Only combine same variables
• 3x + 2x = 5x (correct)
• 3x + 2y stays as is (can’t combine)

3. Factorizing Errors

• Check by expanding back
• Verify factors multiply correctly
• Don’t forget common factors

Related Resources

• IGCSE Maths Past Papers Algebra
• IGCSE Maths Revision Notes
• IGCSE Maths Tutor

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Understanding algebra is essential for IGCSE Maths. Practice regularly and seek help when needed to master this fundamental concept.