Algebra: Polynomials, Linear and Quadratic Equations, AP and Surds
Algebra carries the largest weight in the Odisha BSE HSC paper: zeros of polynomials and their coefficient relations, consistency of linear pairs, solving quadratics by factorisation or formula, arithmetic progressions and surd simplification.
Key results
- For ax² + bx + c with zeros α, β: α + β = −b/a and αβ = c/a. A quadratic from given zeros is k[x² − (sum)x + product].
- Linear pair a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0: unique solution if a₁/a₂ ≠ b₁/b₂; no solution if a₁/a₂ = b₁/b₂ ≠ c₁/c₂; infinitely many if all three ratios are equal.
- Quadratic formula x = (−b ± √(b² − 4ac))/2a with discriminant D = b² − 4ac: D > 0 two distinct real roots, D = 0 equal real roots, D < 0 no real roots.
- AP: aₙ = a + (n − 1)d; Sₙ = n/2[2a + (n − 1)d] = n/2(a + l).
- Rationalise with the conjugate: 1/(√a − √b) = (√a + √b)/(a − b).
Worked example
Solve 2x² − 7x + 3 = 0. D = (−7)² − 4(2)(3) = 49 − 24 = 25 > 0, so two distinct real roots. x = (7 ± 5)/4, giving x = 3 or x = 1/2. Check: sum 3 + 1/2 = 7/2 = −b/a; product 3 × 1/2 = 3/2 = c/a.
Traps that cost marks
- Dropping the minus sign in −b/a or in −b of the formula.
- Using n instead of (n − 1) in the nth-term formula, or n instead of n/2 in the sum.
- Declaring a root without verifying it in the original equation, especially after squaring a surd equation.
- Comparing ratios without first writing both equations with all terms on the left.