Straight lines and quadratic equations

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📐 STRAIGHT LINES & QUADRATIC EQUATIONS

Cambridge IGCSE Mathematics - Quick Reference Guide

PART 1: STRAIGHT LINES 📊

Standard Form

y = mx + c

• m = gradient (slope/steepness)
• c = y-intercept (crosses y-axis)
• x, y = coordinates

Example: y = 3x + 2
• Gradient = 3 (rises 3 for every 1 across)
• Y-intercept = 2 (crosses at (0,2))

Gradient Formula

m = (y₂ - y₁) / (x₂ - x₁)

Type	Value	Direction
Positive	m > 0	↗ Up right
Negative	m < 0	↘ Down right
Zero	m = 0	→ Horizontal
Undefined	-	↑ Vertical

Example: Points (2, 4) and (6, 12)
m = (12 - 4)/(6 - 2) = 8/4 = 2

Special Lines

• Horizontal: y = k (e.g. y = 3)
• Vertical: x = k (e.g. x = -2)

Parallel Lines

Same gradient: m₁ = m₂

Lines y = 2x + 1 and y = 2x - 5 are parallel (both m = 2)

Perpendicular Lines

m₁ × m₂ = -1
or m₂ = -1/m₁

Line 1	Line 2 (⊥)
m = 2	m = -½
m = 3	m = -⅓
m = -4	m = ¼

Intercepts

• Y-intercept: Set x = 0, solve for y → (0, y)
• X-intercept: Set y = 0, solve for x → (x, 0)

Example: y = 6x - 12
• Y-intercept: x=0 → y=-12 → (0, -12)
• X-intercept: y=0 → 6x=12 → x=2 → (2, 0)

Distance Between Points

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Example: (1, 1) to (7, 9)
d = √[(7-1)² + (9-1)²] = √[36 + 64] = √100 = 10

Midpoint

M = ((x₁+x₂)/2, (y₁+y₂)/2)

Example: (3, 4) and (5, 10)
M = ((3+5)/2, (4+10)/2) = (4, 7)

PART 2: QUADRATIC EXPRESSIONS 🔢

Standard Form

ax² + bx + c

Contains x² term (highest power is 2)

Expanding Brackets

(x + a)(x + b) = x² + (a+b)x + ab

FOIL Method:
• First: x × x
• Outside: x × b
• Inside: a × x
• Last: a × b

Example: (x + 3)(x + 5)
= x² + 5x + 3x + 15
= x² + 8x + 15

Special Patterns

Pattern	Formula
Square (positive)	(x + a)² = x² + 2ax + a²
Square (negative)	(x - a)² = x² - 2ax + a²
Difference of squares	x² - a² = (x - a)(x + a)

⚠️ Common Mistake:
❌ (x + 3)² = x² + 9 (WRONG!)
✓ (x + 3)² = x² + 6x + 9 (CORRECT!)

Examples:
• (x + 4)² = x² + 8x + 16
• (x - 5)² = x² - 10x + 25
• x² - 49 = (x - 7)(x + 7)

Factorising Quadratics

x² + bx + c = (x + p)(x + q)

Find two numbers that:
• MULTIPLY to give c
• ADD to give b

Example: x² + 7x + 12
Need: multiply to 12, add to 7
Factor pairs: 1×12, 2×6, 3×4
3 + 4 = 7 ✓
x² + 7x + 12 = (x + 3)(x + 4)

Sign Rules

Expression	Brackets	Example
x² + bx + c	(+)(+)	(x+3)(x+4)
x² - bx + c	(-)(-)	(x-3)(x-5)
x² + bx - c	(+)(-)	(x+5)(x-3)
x² - bx - c	(-)(+)	(x-5)(x+3)

PART 3: SOLVING QUADRATIC EQUATIONS ✅

ax² + bx + c = 0

Zero Product Property

If A × B = 0, then A = 0 OR B = 0

Steps to Solve

1. Rearrange to = 0
2. Factorise the expression
3. Set each factor = 0
4. Solve for x

Example 1: x² - 9x = 0
x(x - 9) = 0
x = 0 or x - 9 = 0
x = 0 or x = 9

Example 2: x² - 7x + 12 = 0
(x - 4)(x - 3) = 0
x - 4 = 0 or x - 3 = 0
x = 4 or x = 3

Example 3: x² + 6x = 16
x² + 6x - 16 = 0
(x + 8)(x - 2) = 0
x = -8 or x = 2

📝 FORMULA SUMMARY

STRAIGHT LINES
y = mx + c
m = (y₂-y₁)/(x₂-x₁)
Parallel: m₁ = m₂
Perpendicular: m₁×m₂ = -1
Distance: d = √[(x₂-x₁)² + (y₂-y₁)²]
Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)

QUADRATICS
(x+a)(x+b) = x² + (a+b)x + ab
(x+a)² = x² + 2ax + a²
(x-a)² = x² - 2ax + a²
x² - a² = (x-a)(x+a)
Solve: Factorise → Set = 0

🎯 KEY REMINDERS

DO:
✓ Use ruler for straight lines
✓ Check gradient signs (+ up, - down)
✓ Verify factorising by expanding
✓ Rearrange to = 0 before solving
✓ Write both solutions for quadratics

DON'T:
❌ Forget middle term when squaring
❌ Mix up parallel/perpendicular
❌ Ignore negative signs
❌ Skip checking your answers

🌟 SUCCESS FORMULA

Practice + Persistence = Mastery
Keep working, you've got this! 🚀

📚 Study Smart • 💪 Practice Hard • 🎯 Succeed!

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