Algebra (Identities, Linear/Quadratic Equations)

Algebra (Identities, Linear/Quadratic Equations)

Concept

Algebra in SSC CGL Tier 2 focuses on three areas: algebraic identities (for quick simplification), linear equations (solved by substitution or elimination), and quadratic equations (solved by factorisation or quadratic formula). The identities are the real speed booster — questions that look complex often collapse using the right identity.

Key Identities to Master: (a+b)², (a−b)², a²−b², (a+b+c)² = a²+b²+c²+2(ab+bc+ca), (a+b)³ = a³+3a²b+3ab²+b³, a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca).

Linear Equations in Two Variables: ax + by = c and dx + ey = f. Solve by elimination or substitution. Consistency condition: a/e ≠ b/d for unique solution; a/e = b/d = c/f for infinitely many solutions; a/e = b/d ≠ c/f for no solution.

Quadratic Equations: Convert to standard form ax² + bx + c = 0. Solve by: (1) factorisation if roots are integers/rationals, (2) quadratic formula if not. The discriminant D = b² − 4ac determines nature: D > 0 → two distinct real roots; D = 0 → equal roots; D < 0 → no real roots.

Key Points

• If roots are α, β: α+β = −b/a, αβ = c/a. This is Vieta’s formula.
• If one root is 2+√3, the other is 2−√3 (conjugate pair for rational coefficients).
• For linear equation substitution: if x/a + y/b = 1 and x/c + y/d = 1, solve by cross-multiplication.
• Always rearrange to standard form before identifying a, b, c for quadratic formula.
• (x + 1/x)² = x² + 2 + 1/x² — useful for problems giving x + 1/x and asking for x² + 1/x².

Worked Example

Q: If x + 1/x = 3, find x² + 1/x². Approach: (x + 1/x)² = x² + 2 + 1/x² = 3² = 9. So x² + 1/x² = 9 − 2 = 7. Answer: 7

SSC Pattern / Tips

• Quadratic equations in SSC often have integer or simple fractional roots — try factorisation first.
• For identity-based simplification, always look for patterns: (x+y)² − (x−y)² = 4xy.
• When setting up linear equations from word problems, identify two conditions to form two equations.
• If quadratic has complex roots, they come in conjugate pairs — this doesn’t affect sum/product calculations.