Straight lines and quadratic equations C

📐 Straight Lines & Quadratic Equations

Complete Formulas - Grade 9 Cambridge IGCSE

Part 1: STRAIGHT LINES

1.1 Equation of a Line

y = mx + c

m = gradient (slope)
c = y-intercept

1.2 Gradient (Slope)

m = (y₂ - y₁)/(x₂ - x₁)

m = rise/run

1.3 Special Lines

Type	Equation	Gradient
Horizontal	y = c	0
Vertical	x = a	Undefined

1.4 Parallel Lines

Parallel: m₁ = m₂

1.5 Perpendicular Lines

Perpendicular: m₁ × m₂ = -1
Or: m₂ = -1/m₁

1.6 Distance Formula

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

1.7 Midpoint Formula

M = ((x₁+x₂)/2, (y₁+y₂)/2)

Example: Midpoint of (2,8) and (10,4)
M = ((2+10)/2, (8+4)/2) = (6, 6)

Part 2: QUADRATICS

2.1 Expanding Brackets

(x + a)(x + b) = x² + (a+b)x + ab

2.2 FOIL Method

First × First
Outside × Outside
Inside × Inside
Last × Last

(x + 3)(x + 5)
F: x·x = x²
O: x·5 = 5x
I: 3·x = 3x
L: 3·5 = 15
= x² + 8x + 15

2.3 Perfect Squares

(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²

❌ (x+4)² ≠ x²+16
✅ (x+4)² = x²+8x+16

2.4 Difference of Two Squares

a² - b² = (a+b)(a-b)

Expression	Factorised
x² - 49	(x+7)(x-7)
x² - 1	(x+1)(x-1)
4x² - 9	(2x+3)(2x-3)

2.5 Factorising Quadratics

x² + bx + c = (x + p)(x + q)

Find p,q where:
p × q = c
p + q = b

Factorise: x² + 7x + 12

Need: p×q = 12, p+q = 7
Try: 3×4 = 12, 3+4 = 7 ✓
Answer: (x+3)(x+4)

2.6 Solving Quadratic Equations

If a × b = 0, then a = 0 OR b = 0

Steps:
1️⃣ Rearrange to = 0
2️⃣ Factorise
3️⃣ Set each bracket = 0
4️⃣ Solve

Solve: x² - 7x + 12 = 0

Factorise: (x-4)(x-3) = 0
x-4=0 or x-3=0
x = 4 or x = 3

📋 Quick Reference

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

QUADRATICS:
• (x+a)(x+b) = x²+(a+b)x+ab
• (a±b)² = a²±2ab+b²
• a²-b² = (a+b)(a-b)

Practice makes perfect! 📐✨