Quadratic Equations and Arithmetic Progressions
This section covers Quadratic Equations and Arithmetic Progressions. The exam tests the discriminant, roots by factorisation or the quadratic formula, and the nth term and sum of an AP.
Quadratic equations
For ax² + bx + c = 0 with a ≠ 0, the discriminant is D = b² - 4ac.
- D > 0: two distinct real roots.
- D = 0: two equal real roots.
- D < 0: no real roots. The roots are x = (-b ± √D)/(2a); their sum is -b/a and product is c/a.
Arithmetic progressions
For first term a and common difference d:
- nth term: aₙ = a + (n - 1)d.
- Sum of n terms: Sₙ = n/2[2a + (n - 1)d] = n/2(a + l), where l is the last term.
- Three terms in AP can be taken as a - d, a, a + d.
Worked example
Solve x² - 5x + 6 = 0. D = 25 - 24 = 1, so the roots are (5 ± 1)/2, giving 3 and 2. For the AP 7, 10, 13, …, a = 7 and d = 3. The 15th term is 7 + 14×3 = 49. The sum of the first 15 terms is 15/2(7 + 49) = 15×28 = 420.
Traps
A negative discriminant means no real roots, not “no roots”. In an AP, n must be a positive integer; a fractional n signals a wrong equation. Use the last term formula only when the last term is known, and keep the common difference sign when the AP decreases.