Algebra: quadratic equations, surds and variation
Algebra (WBBSE Madhyamik)
Algebra carries quadratic equations in one variable, quadratic surds, and variation. Questions appear as direct solves, as word problems that must first be modelled into an equation, and as simplifications requiring rationalisation.
Core rules
- Standard form ax² + bx + c = 0 with a ≠ 0. Sridhar Acharya's formula gives x = (−b ± √(b² − 4ac))/(2a).
- The discriminant D = b² − 4ac settles the nature of roots: D > 0 gives two distinct real roots, D = 0 gives equal roots, D < 0 gives no real root.
- Sum of roots = −b/a; product of roots = c/a.
- Completing the square rewrites ax² + bx + c as a(x + b/2a)² + (c − b²/4a).
- Surds: √a × √b = √(ab); rationalise using the conjugate, since (a + √b)(a − √b) = a² − b.
- Direct variation y ∝ x gives y = kx; inverse variation y ∝ 1/x gives xy = k.
Worked example
Solve 2x² − 7x + 3 = 0. Here a = 2, b = −7, c = 3, so D = 49 − 24 = 25 > 0. Then x = (7 ± 5)/4, giving x = 3 or x = 1/2. Check: sum = 3.5 = 7/2 = −b/a and product = 1.5 = 3/2 = c/a, both correct.
Traps
- Dropping the ± and reporting only one root.
- Sign slips when b is negative, because −b turns into a positive number.
- Cancelling across a sum, for example writing (x² + 4)/x as x + 4.
- Modelling a problem as direct variation when the wording says 'varies inversely'.
- Keeping a root that makes a denominator zero in a word problem.