Factorisation Calculator
Factorise quadratics and special expressions with step-by-step solutions. Learn the methods or get quick answers.
Example: For x² + 5x + 6, enter a=1, b=5, c=6
Key Formulas
a² - b² = (a + b)(a - b)
x² - 9 = (x + 3)(x - 3)
a² + 2ab + b² = (a + b)²
x² + 6x + 9 = (x + 3)²
a² - 2ab + b² = (a - b)²
x² - 6x + 9 = (x - 3)²
a³ + b³ = (a + b)(a² - ab + b²)
x³ + 8 = (x + 2)(x² - 2x + 4)
Common Mistakes
Forgetting to check for common factors
Always look for HCF first: 2x² + 4x = 2x(x + 2)
Wrong signs when factorising
If c is positive, both signs are the same. If c is negative, signs are different.
Confusing sum/difference of squares
a² + b² CANNOT be factorised with real numbers!
Understanding Factorisation
What is Factorisation?
Factorisation is the process of writing an expression as a product of its factors. It's the opposite of expanding brackets.
Example:
x² + 5x + 6 = (x + 2)(x + 3)
Why is it Important?
- •Solving quadratic equations
- •Simplifying algebraic fractions
- •Finding x-intercepts of graphs
- •Essential for A-Level calculus
Factorisation Methods
1. Simple Quadratics (when a = 1)
For x² + bx + c, find two numbers that add to b and multiply to c.
x² + 7x + 12 = (x + 3)(x + 4)
Because 3 + 4 = 7 and 3 × 4 = 12
2. AC Method (when a ≠ 1)
For ax² + bx + c, find two numbers that add to b and multiply to a × c. Then split the middle term and factor by grouping.
2x² + 7x + 3 = (2x + 1)(x + 3)
a × c = 6. Numbers: 1 and 6 (add to 7, multiply to 6)
3. Difference of Two Squares
When you have a² - b², use the formula: a² - b² = (a + b)(a - b)
x² - 25 = (x + 5)(x - 5)
Note: a² + b² cannot be factorised with real numbers!
4. Sum & Difference of Cubes
Use SOAP: Same, Opposite, Always Positive
Sum of Cubes:
a³ + b³ = (a + b)(a² - ab + b²)
Difference of Cubes:
a³ - b³ = (a - b)(a² + ab + b²)
Quick Reference
Perfect Squares
1
1²
4
2²
9
3²
16
4²
25
5²
36
6²
49
7²
64
8²
81
9²
100
10²
Perfect Cubes
1
1³
8
2³
27
3³
64
4³
125
5³
216
6³
343
7³
512
8³
729
9³
1000
10³
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