Linear Equations and Quadratic Equations
Linear Equations and Quadratic Equations covers simultaneous linear systems, consistency, the quadratic formula, the discriminant and the nature of roots. The exam often links algebra to a short word problem.
Core rules
- A pair a₁x + b₁y = c₁, a₂x + b₂y = c₂ has a unique solution when a₁/a₂ ≠ b₁/b₂, no solution when a₁/a₂ = b₁/b₂ ≠ c₁/c₂, and infinitely many when all three ratios are equal.
- Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a).
- Discriminant D = b² − 4ac: D > 0 gives two distinct real roots; D = 0 gives equal real roots; D < 0 gives no real roots.
- For ax² + bx + c = 0, sum of roots = −b/a and product = c/a.
Worked example
Solve 2x² − 7x + 3 = 0. Here D = 49 − 24 = 25, so the roots are real and distinct. x = [7 ± 5]/4, giving x = 3 or x = 1/2. Check the product: 3 × 1/2 = 3/2 = c/a, which confirms the values.
Traps
In the formula, −b means the whole coefficient changes sign; for b = −7 the numerator starts with +7. Do not cancel 2 from only part of a fraction. In word problems, reject a root that makes a length or age negative even if it satisfies the equation. For simultaneous equations, substituting back into the original equation catches arithmetic errors.