1. Standard form
ax² + bx + c = 0 (a ≠ 0). Every quadratic has two roots α, β (real or not).
2. Finding roots
- Factorisation: split b into two numbers whose product is ac. x² − 7x + 12 = (x − 3)(x − 4) → 3, 4.
- Formula: x = [−b ± √(b² − 4ac)]/2a.
3. Sum and product
α + β = −b/a, αβ = c/a. Equation from roots: x² − (α + β)x + αβ = 0.
| Expression | In terms of sum S and product P |
|---|---|
| α² + β² | S² − 2P |
| (α − β)² | S² − 4P |
| 1/α + 1/β | S/P |
| α³ + β³ | S³ − 3PS |
| α/β + β/α | (S² − 2P)/P |
4. Nature of roots — discriminant D = b² − 4ac
| D | Roots |
|---|---|
| > 0 | real and distinct (rational if D is a perfect square) |
| = 0 | real and equal (= −b/2a) |
| < 0 | no real roots |
5. Solved examples
(ASO 2021) α + β = 12, α − β = 4 → 8 and 4 → x² − 12x + 32 = 0.
(ASO 2021) Discriminant of 3x² − 5x + 3 = 25 − 36 = −11 → no real roots.
(ASO 2015) 2x² − 2√2x + 1 = 0 → (√2x − 1)² = 0 → x = 1/√2, 1/√2.
kx² + 6x + 1 = 0 has equal roots → 36 − 4k = 0 → k = 9.
Roots of x² − 5x + 6 = 0 are α, β → α² + β² = 25 − 12 = 13.
One root of x² + px + 12 = 0 is 4 → 16 + 4p + 12 = 0 → p = −7; other root = 12/4 = 3.
6. Word problems
Product of two consecutive positive integers is 306 → n(n + 1) = 306 → n = 17 → 17 and 18.
A rectangle's length is 3 m more than its breadth and area 108 m² → b(b + 3) = 108 → b = 9.
Solved examples by question type
Each type that comes in the ASO paper, worked step by step. Try it yourself first, then read the solution.
Basic level
Example 1. The roots of x² − 2x − 120 = 0 are:
(a) 2, -120 (b) -9, 13 (c) -12, 10 (d) -10, 12
Answer: (d) -10, 12
Solution: (x − -10)(x − 12) = 0 → x = -10, 12.
Example 2. The roots of x² − 13x + 30 = 0 are:
(a) 4, 11 (b) -10, -3 (c) 13, 30 (d) 3, 10
Answer: (d) 3, 10
Solution: (x − 10)(x − 3) = 0 → x = 10, 3.
ASO exam level
Example 3. The roots of x² − 4x − 21 = 0 are:
(a) -21 and 4 (b) -3 and 7 (c) -2 and 6 (d) 3 and -7
Answer: (b) -3 and 7
Solution: Find two numbers with sum 4 and product -21: -3 and 7.
Example 4. If α and β are roots of a quadratic equation with α + β = 8 and α − β = 6, the equation is:
(a) x² − 6x + 7 = 0 (b) x² + 8x + 7 = 0 (c) x² − 8x − 7 = 0 (d) x² − 8x + 7 = 0
Answer: (d) x² − 8x + 7 = 0
Solution: α = 7, β = 1. Equation: x² − (α+β)x + αβ = x² − 8x + 7 = 0.
Example 5. The equation ax² + bx + c = 0 has real and distinct roots if:
(a) b² − 4ac < 0 (b) b² − 4ac > 0 (c) b² − 4ac = 0 (d) b² + 4ac > 0
Answer: (b) b² − 4ac > 0
Solution: Discriminant D = b² − 4ac. D > 0: real and distinct; D = 0: equal; D < 0: not real.
Example 6. The equation ax² + bx + c = 0 has real and distinct roots if:
(a) b² − 4ac = 0 (b) b² + 4ac > 0 (c) b² − 4ac > 0 (d) b² − 4ac < 0
Answer: (c) b² − 4ac > 0
Solution: Discriminant D = b² − 4ac. D > 0: real and distinct; D = 0: equal; D < 0: not real.
Example 7. For what positive value of k does 5x² + kx + 5 = 0 have equal roots?
(a) 100 (b) 11 (c) 12 (d) 10
Answer: (d) 10
Solution: Equal roots: k² = 4 × 5 × 5 = 100 → k = 10.
Example 8. For what positive value of k does 1x² + kx + 1 = 0 have equal roots?
(a) 2 (b) 3 (c) 4 (d) 0
Answer: (a) 2
Solution: Equal roots: k² = 4 × 1 × 1 = 4 → k = 2.
Example 9. The quadratic equation whose roots are 4 and -2 is:
(a) x² − 2x + 8 = 0 (b) x² + 8x + 2 = 0 (c) x² + 2x − 8 = 0 (d) x² − 2x − 8 = 0
Answer: (d) x² − 2x − 8 = 0
Solution: x² − (sum)x + product: sum = 2, product = -8.
Example 10. The nature of the roots of 9x² + 4x − 5 = 0 is:
(a) Real and equal (b) Rational and equal (c) Real and distinct (d) No real roots
Answer: (c) Real and distinct
Solution: D = b² − 4ac = 16 − -180 = 196 → real and distinct.
Advanced level
Example 11. If α and β are the roots of x² − 1x − 56 = 0, then α² + β² = ?
(a) 113 (b) -111 (c) 57 (d) 1
Answer: (a) 113
Solution: α² + β² = (α + β)² − 2αβ = 1 − -112 = 113.
Example 12. If α and β are the roots of x² + 4x + 0 = 0, then α² + β² = ?
(a) 19 (b) 17 (c) 16 (d) 14
Answer: (c) 16
Solution: α² + β² = (α + β)² − 2αβ = 16 − 0 = 16.
Topic-wise practice for Quadratic equations (Quant): Roots, discriminant, forming equations. Useful for OPSC ASO, OSSC, OSSSC and other Odisha exams.
Question types covered
- Roots by factorisation
- Roots by the quadratic formula
- Discriminant and nature of roots
- Sum and product of roots
- Forming the equation from its roots
- Word problems on quadratic equations