Factorizing Simple Quadratics
GCSE Mathematics📐 What are Simple Quadratics?
Simple quadratics are expressions of the form x² + bx + c where the coefficient of x² is 1. Factorizing means writing it as a product of two brackets: (x + p)(x + q) where p + q = b and p × q = c.
where: p + q = b and p × q = c
Example: x² + 7x + 12 = (x + 3)(x + 4)
Check: 3 + 4 = 7 ✓, 3 × 4 = 12 ✓
🎨 The Number Pair Method
📚 Types of Simple Quadratics
➕ Both numbers positive
When b and c are both positive, both p and q are positive.
Example 1: x² + 5x + 6
Example 2: x² + 7x + 12
Example 3: x² + 8x + 15
➖ One number negative
When c is negative, one factor is positive and one is negative.
Example 1: x² + 2x - 15
Example 2: x² - 4x - 12
Example 3: x² - x - 20
➖➖ Both numbers negative
When b is negative and c is positive, both numbers are negative.
Example 1: x² - 7x + 12
Example 2: x² - 8x + 15
Example 3: x² - 9x + 20
🔄 Difference of Two Squares
Special case: x² - a² = (x - a)(x + a)
Example 1: x² - 9
Example 2: x² - 25
Example 3: x² - 16
📊 Common Factor Pairs
| c (constant term) | Factor Pairs | Sum (b) |
|---|---|---|
| 6 | (1,6), (2,3) | 7, 5 |
| 12 | (1,12), (2,6), (3,4) | 13, 8, 7 |
| 20 | (1,20), (2,10), (4,5) | 21, 12, 9 |
| -6 | (-1,6), (1,-6), (-2,3), (2,-3) | 5, -5, 1, -1 |
🧮 Interactive Quadratic Factorizer
Enter a quadratic and see how it factorizes step-by-step!
✏️ Practice Factorizing Quadratics
Factorize: x² + 7x + 12
⚠️ Common Mistakes
❌ Wrong: Mixing up signs
x² - 5x + 6 = (x - 2)(x - 3) ✓
x² - 5x + 6 = (x + 2)(x + 3) ❌ (gives +5x)
✅ Correct: Check signs carefully
If c positive and b negative → both factors negative
If c negative → one factor positive, one negative
❌ Wrong: Wrong factor pair
x² + 7x + 12 = (x + 2)(x + 6) ❌ (2×6=12 but 2+6=8, not 7)
Correct: (x + 3)(x + 4) ✓
⚠️ Forgetting the difference of squares
x² - 16 = (x - 4)(x + 4), not (x - 4)²
🌍 Where Quadratics Appear
📐 Area Problems
A rectangle has area x² + 5x + 6
Length = x + 3, Width = x + 2
Area = (x+3)(x+2) = x² + 5x + 6
🎯 Projectile Motion
Height = -5t² + 20t + 25
Factor to find when height = 0
💰 Profit Maximization
Profit = -x² + 100x - 2400
Factor to find break-even points