Algebra Cheat Sheet

Algebra Cheat Sheet

Quick Revision

Algebra uses letters for unknown numbers. Most exam questions are solved quickly if you know the identities and the quadratic formulas by heart.

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1. Basic Identities

(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
a² − b² = (a + b)(a − b)
(a + b)² + (a − b)² = 2(a² + b²)
(a + b)² − (a − b)² = 4ab

2. Cubic Identities

(a + b)³ = a³ + 3a²b + 3ab² + b³
(a − b)³ = a³ − 3a²b + 3ab² − b³
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)

3. Three-Variable Identities

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

Special case: if a + b + c = 0, then a³ + b³ + c³ = 3abc.

4. Laws of Indices

aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
a⁰ = 1   (a ≠ 0)
a⁻ⁿ = 1 / aⁿ

5. Quadratic Equations

For ax² + bx + c = 0 (a ≠ 0):

Roots = [ −b ± √(b² − 4ac) ] / 2a
Discriminant  D = b² − 4ac
Sum of roots      = −b / a
Product of roots  = c / a

Nature of roots: D > 0 → real and distinct; D = 0 → real and equal; D < 0 → imaginary.

6. Exam Quick Tips

• Memorise a² − b² = (a+b)(a−b) and the cube identities — they shortcut many sums.
• If a + b + c = 0, jump straight to a³ + b³ + c³ = 3abc.
• For a quadratic, read off sum = −b/a and product = c/a without solving.
• The discriminant tells you the nature of the roots before you compute them.

Keep revising with the rest of the Maths section:

• All Maths topics
• Simplification Cheat Sheet
• Fractions and Decimals Cheat Sheet

How this was made: Drafted with AI assistance; spot-checked by the editor on 31 October 2026.