Algebra (RRB) — Core

Linear/quadratic equations, identities, sequences.

Linear, quadratic & polynomial essentials

Notes

Linear equations in one variable: ax + b = 0 ⟹ x = −b/a.

Two linear equations (two variables): use elimination or substitution.

Elimination: multiply equations to match coefficients of one variable, then subtract.
Cross-multiplication shortcut for ax + by + c = 0; a'x + b'y + c' = 0:
x / (bc' − b'c) = y / (a'c − ac') = 1 / (ab' − a'b)

Quadratic equations: ax² + bx + c = 0 (a ≠ 0).

Discriminant: D = b² − 4ac.
- D > 0 → two distinct real roots.
- D = 0 → equal real roots (repeated).
- D < 0 → no real roots (complex).
Roots: x = (−b ± √D) / 2a.
Sum of roots = −b/a; product of roots = c/a.

Useful identities:

(a+b)² = a² + 2ab + b²
(a−b)² = a² − 2ab + b²
a² − b² = (a+b)(a−b)
(a+b)³ = a³ + 3a²b + 3ab² + b³
a³ + b³ = (a+b)(a² − ab + b²)
a³ − b³ = (a−b)(a² + ab + b²)
a³ + b³ + c³ − 3abc = (a+b+c)(a² + b² + c² − ab − bc − ca)
If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Indices laws:

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

Worked examples — equations & expressions

Worked example

Example 1 — Two-variable:
Solve: 2x + 3y = 13; 4x − y = 5.
Method: From the 2nd, y = 4x − 5. Substitute: 2x + 3(4x−5) = 13 ⟹ 2x + 12x − 15 = 13 ⟹ 14x = 28 ⟹ x = 2, y = 3.

Example 2 — Quadratic via sum/product:
If sum of roots = 7 and product = 12, find the quadratic.
Method: x² − (sum)x + (product) = 0 ⟹ x² − 7x + 12 = 0 ⟹ (x−3)(x−4) = 0. So roots are 3 and 4.

Example 3 — Identity shortcut:
If a + b = 5 and ab = 6, find a³ + b³.
Method: a³ + b³ = (a+b)³ − 3ab(a+b) = 125 − 3·6·5 = 125 − 90 = 35.

Example 4 — a + 1/a pattern:
If x + 1/x = 4, find x² + 1/x².
Method: (x + 1/x)² = x² + 2 + 1/x². So 16 = x² + 1/x² + 2 ⟹ x² + 1/x² = 14.

For x³ + 1/x³ use: x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x) = 64 − 12 = 52.

Example 5 — Discriminant:
Find k so that 2x² + kx + 8 = 0 has equal roots.
Method: D = 0 ⟹ k² − 64 = 0 ⟹ k = ±8.

Example 6 — Three-variable identity (often asked):
If a + b + c = 0, find (a³ + b³ + c³)/(abc).
Method: By identity, a³ + b³ + c³ = 3abc when a+b+c=0. Answer: 3.

Speed tip: spot the pattern (x + 1/x form, sum-product, a³+b³ identity) before reaching for algebra. RRB algebra is mostly recognition, not heavy computation.