Standard Identities
| Identity | Use |
|---|---|
| (a + b)² = a² + 2ab + b² | 103² = 10,000 + 600 + 9 = 10,609 |
| (a − b)² = a² − 2ab + b² | 98² = 10,000 − 400 + 4 = 9604 |
| a² − b² = (a + b)(a − b) | 57² − 43² = 100 × 14 = 1400 |
| (a + b)³ = a³ + b³ + 3ab(a + b) | a³ + b³ = (a + b)³ − 3ab(a + b) |
| (a − b)³ = a³ − b³ − 3ab(a − b) | a³ − b³ = (a − b)³ + 3ab(a − b) |
| a³ + b³ = (a + b)(a² − ab + b²) | Factorising sums of cubes |
| (a + b + c)² = a² + b² + c² + 2(ab + bc + ca) | Sum of squares from a sum |
| a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca) | If a + b + c = 0, then a³ + b³ + c³ = 3abc |
x + 1/x results
If x + 1/x = k, then x² + 1/x² = k² − 2 and x³ + 1/x³ = k³ − 3k. With k = 3: x² + 1/x² = 7 and x³ + 1/x³ = 18. If x − 1/x = k, then x² + 1/x² = k² + 2.
Linear Equations
- One variable: do the same operation on both sides. 3x − 7 = 2x + 5 gives x = 12.
- Two variables: eliminate one variable by adding or subtracting multiples. x + y = 10 and x − y = 4 give 2x = 14, so x = 7 and y = 3.
- For a₁x + b₁y = c₁ and a₂x + b₂y = c₂: a unique solution if a₁/a₂ ≠ b₁/b₂; no solution (parallel lines) if a₁/a₂ = b₁/b₂ ≠ c₁/c₂; infinitely many if all three ratios are equal.
Quadratic Equations
Roots of ax² + bx + c = 0
x = [−b ± √(b² − 4ac)] / 2a
Sum of roots = −b/a and product = c/a. x² − 5x + 6 = 0 has roots 2 and 3 (sum 5, product 6).
| Discriminant D = b² − 4ac | Roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | Two equal real roots, each −b/2a |
| D < 0 | No real roots |
A quadratic with roots α and β is x² − (α + β)x + αβ = 0. Useful forms: α² + β² = (α + β)² − 2αβ and 1/α + 1/β = (α + β)/αβ.
Polynomials
- Remainder theorem: when p(x) is divided by (x − a), the remainder is p(a). x³ − 2x + 1 divided by (x − 2) leaves 8 − 4 + 1 = 5.
- Factor theorem: (x − a) is a factor of p(x) exactly when p(a) = 0.
- The degree is the highest power. A polynomial of degree n has at most n roots.
Common Mistakes
Writing (a + b)² as a² + b²; taking the sum of roots as +b/a instead of −b/a; and dividing both sides by a variable that could be zero, which loses a root (x² = 3x has roots 0 and 3).