60-second revision
- x + 1/x = k gives x² + 1/x² = k² − 2 and x³ + 1/x³ = k³ − 3k.
- x − 1/x = k gives x² + 1/x² = k² + 2 and x³ − 1/x³ = k³ + 3k.
- If a + b + c = 0, then a³ + b³ + c³ = 3abc.
- a³ + b³ = (a + b)³ − 3ab(a + b).
- ax + by = c cuts the axes at (c/a, 0) and (0, c/b).
- To solve index equations, write both sides with the same base and equate powers.
- Rationalise by multiplying with the conjugate: (sqrt(a) + sqrt(b))(sqrt(a) − sqrt(b)) = a − b.
Identities to know cold
| Identity | Expanded form |
|---|---|
| (a + b)² | a² + 2ab + b² |
| (a − b)² | a² − 2ab + b² |
| a² − b² | (a + b)(a − b) |
| (a + b)³ | a³ + b³ + 3ab(a + b) |
| (a − b)³ | a³ − b³ − 3ab(a − b) |
| a³ + b³ | (a + b)(a² − ab + b²) |
| a³ − b³ | (a − b)(a² + ab + b²) |
| (a + b + c)² | a² + b² + c² + 2(ab + bc + ca) |
| a³ + b³ + c³ − 3abc | (a + b + c)(a² + b² + c² − ab − bc − ca) |
Worked example: if a − b = 4 and ab = 5, then a² + b² = (a − b)² + 2ab = 16 + 10 = 26, and a³ − b³ = (a − b)³ + 3ab(a − b) = 64 + 60 = 124.
The x + 1/x family
- x + 1/x = k: x² + 1/x² = k² − 2, x³ + 1/x³ = k³ − 3k.
- x − 1/x = k: x² + 1/x² = k² + 2, x³ − 1/x³ = k³ + 3k.
- x^4 + 1/x^4 = (x² + 1/x²)² − 2.
Worked example: x + 1/x = 4. Then x² + 1/x² = 14, x³ + 1/x³ = 64 − 12 = 52, and x^4 + 1/x^4 = 196 − 2 = 194.
When a + b + c = 0
Then a³ + b³ + c³ = 3abc. Example: 7, −3, −4 add to 0, so 343 − 27 − 64 = 252 = 3 × 7 × (−3) × (−4). The same fact gives (x − y)³ + (y − z)³ + (z − x)³ = 3(x − y)(y − z)(z − x).
Linear equations
One variable: expand, collect x terms on one side, divide. Two variables: substitute or eliminate.
Worked example: 4x − 3y = 1 and 2x + y = 13. From the second, y = 13 − 2x. Substituting, 4x − 39 + 6x = 1, so x = 4 and y = 5.
| For a1x + b1y = c1 and a2x + b2y = c2 | Solutions | Lines are |
|---|---|---|
| a1/a2 not equal to b1/b2 | Exactly one | Intersecting |
| a1/a2 = b1/b2, but not equal to c1/c2 | None | Parallel |
| a1/a2 = b1/b2 = c1/c2 | Infinitely many | Coincident |
Linear graphs
- ax + by = c meets the x-axis at (c/a, 0) and the y-axis at (0, c/b).
- Area of the triangle it makes with the axes = 1/2 × |c/a| × |c/b|.
- In y = mx + c, m is the slope and c is the y-intercept; parallel lines share the same slope.
- x = k is a vertical line, y = k a horizontal line; the x-axis is y = 0.
Worked example: 5x + 2y = 20 meets the axes at (4, 0) and (0, 10); the triangle has area 1/2 × 4 × 10 = 20 square units.
Indices
| Law | Example |
|---|---|
| a^m × a^n = a^(m + n) | 2^3 × 2^4 = 2^7 |
| a^m ÷ a^n = a^(m − n) | 5^6 ÷ 5^4 = 25 |
| (a^m)^n = a^(mn) | (3^2)^3 = 3^6 |
| a^0 = 1, a^(−n) = 1/a^n | 4^(−2) = 1/16 |
| a^(1/n) = nth root of a | 27^(1/3) = 3 |
Worked example: 4^(x + 1) = 8^x. Write both in base 2: 2^(2x + 2) = 2^(3x), so x = 2.
Surds
Simplify by pulling out perfect squares: sqrt(72) = sqrt(36 × 2) = 6 sqrt(2). Only like surds add: 3 sqrt(5) + 2 sqrt(5) = 5 sqrt(5). To rationalise, multiply by the conjugate: 1/(sqrt(3) + sqrt(2)) = (sqrt(3) − sqrt(2))/(3 − 2) = sqrt(3) − sqrt(2).
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