Quadratic Equations
Key Concepts
| # | Concept | Explanation |
|---|---|---|
| 1 | Standard form | ax² + bx + c = 0 (a ≠ 0) |
| 2 | Roots formula | x = [-b ± √(b²–4ac)] / 2a |
| 3 | Discriminant (D) | D = b² – 4ac; decides nature of roots |
| 4 | Sum of roots (α+β) | –b / a |
| 5 | Product of roots (αβ) | c / a |
| 6 | Real & equal roots | D = 0 |
| 7 | Rational roots | D is perfect square & a,b,c rational |
| 8 | Word-procedure | Frame equation → simplify → solve → check feasibility |
15 Practice MCQs
1. The roots of x² – 7x + 12 = 0 are
A) 3, 4 B) –3, –4 C) 2, 6 D) 1, 12 Answer: A) 3, 4 Solution: x² – 7x + 12 = (x – 3)(x – 4) = 0 ⇒ x = 3 or 4 Shortcut: factors of 12 that sum to 7 → 3 & 4 Tag: Factorisation2. If 2x² – 5x + k = 0 has equal roots, k = ?
A) 25/4 B) 25/8 C) 5/2 D) 5 Answer: B) 25/8 Solution: D = 0 ⇒ 25 – 8k = 0 ⇒ k = 25/8 Tag: Discriminant3. Sum of roots of 3x² – 12x + 9 = 0 is
A) 4 B) –4 C) 3 D) –3 Answer: A) 4 Solution: –b/a = 12/3 = 4 Tag: Sum of roots4. Product of roots of 5x² + 8x – 3 = 0 is
A) –3/5 B) 8/5 C) 3/5 D) –8/5 Answer: A) –3/5 Solution: c/a = –3/5 Tag: Product of roots5. Which quadratic has irrational roots?
A) x² – 5x + 6 B) x² – 3x + 1 C) x² – 4x + 4 D) x² – 9 Answer: B) x² – 3x + 1 Solution: D = 9 – 4 = 5 (not perfect square) Tag: Nature of roots6. If α, β are roots of x² – 6x + 2 = 0, find α² + β²
A) 32 B) 36 C) 28 D) 30 Answer: A) 32 Solution: α²+β² = (α+β)² – 2αβ = 36 – 4 = 32 Shortcut: memorise identity Tag: Symmetric roots7. One root of x² – (k+4)x + 4k = 0 is 4; the other is
A) k B) 4 C) 1 D) 2 Answer: A) k Solution: product 4k; one factor 4 ⇒ other = k Tag: Product relation8. For what k does (k+1)x² – 4kx + 4 = 0 have real roots?
A) k ≥ 1 B) k ≤ 1 C) k ≥ –1 D) all k Answer: A) k ≥ 1 Solution: D ≥ 0 ⇒ 16k² – 16(k+1) ≥ 0 ⇒ k² – k – 1 ≥ 0 ⇒ k ≥ 1 Tag: Inequality with D9. The equation whose roots are 2+√3 and 2–√3 is
A) x² – 4x + 1 = 0 B) x² + 4x + 1 = 0 C) x² – 4x – 1 = 0 D) x² – 1 = 0 Answer: A) x² – 4x + 1 = 0 Shortcut: sum 4, product 1 ⇒ x² – 4x + 1 = 0 Tag: Form equation from roots10. A train travels 180 km. If speed were 5 km/h more, time taken would be 1 h less. Original speed?
A) 30 km/h B) 36 km/h C) 40 km/h D) 45 km/h Answer: C) 40 km/h Solution: 180/s – 180/(s+5) = 1 ⇒ s² + 5s – 900 = 0 ⇒ s = 40 Tag: Speed-time equation11. If x = 1 is root of ax² – 3x + 2 = 0, then a =
A) 1 B) 2 C) –1 D) 0 Answer: A) 1 Solution: put x = 1 ⇒ a – 3 + 2 = 0 ⇒ a = 1 Tag: Substitution12. The quadratic with rational coefficients whose one root is 3+√2 is
A) x² – 6x + 7 = 0 B) x² – 6x – 7 = 0 C) x² + 6x + 7 = 0 D) x² – 9 = 0 Answer: A) x² – 6x + 7 = 0 Solution: other root 3–√2; sum 6, product 9–2=7 Tag: Conjugate root13. If α, β are roots of 2x² – 3x – 5 = 0, find 1/α + 1/β
A) –3/5 B) 3/5 C) 5/3 D) –5/3 Answer: A) –3/5 Solution: (α+β)/αβ = (3/2)/(–5/2) = –3/5 Tag: Reciprocal sum14. The value of x² – 4x + 9 at x = 2 + i√5 is
A) 0 B) 5 C) 10 D) –5 Answer: A) 0 Solution: x = 2 + i√5 satisfies x² – 4x + 9 = 0 Tag: Complex root verification15. How many real roots for x⁴ – 5x² + 4 = 0?
A) 4 B) 2 C) 1 D) 0 Answer: A) 4 Solution: Let y = x² ⇒ y² – 5y + 4 = 0 ⇒ y = 1, 4 ⇒ x = ±1, ±2 Tag: BiquadraticSpeed Tricks
| Situation | Shortcut | Example |
|---|---|---|
| 1. Sum & product known | Write x² – (sum)x + product = 0 directly | roots 7, –3 ⇒ x² – 4x – 21 = 0 |
| 2. D ends with 2,3,7,8 | No perfect square ⇒ roots irrational | D = 47 → irrational |
| 3. Coefficients a+b+c = 0 | One root is 1, other c/a | 3x² – 5x + 2 = 0 → roots 1, 2/3 |
| 4. Missing bx (b = 0) | x = ±√(–c/a) | 4x² – 9 = 0 → x = ±3/2 |
| 5. Replace x by 1/x | New equation: reverse coefficient order | x² – 5x + 6 = 0 → 6x² – 5x + 1 = 0 |
Quick Revision
| Point | Detail |
|---|---|
| 1 | Always write equation in standard form before using formula. |
| 2 | D > 0 → two real distinct; D = 0 → equal; D < 0 → complex. |
| 3 | If question asks “possible value”, prefer factorisation over formula to save 30 s. |
| 4 | For word problems, check negative/imaginary solutions—reject if speed, length, time negative. |
| 5 | Remember identities: α²+β² = (α+β)² – 2αβ; α³+β³ = (α+β)³ – 3αβ(α+β). |
| 6 | Conjugate surd root theorem: irrational roots occur in pairs for rational coefficients. |
| 7 | Graph of ax²+bx+c opens upwards if a > 0, downwards if a < 0. |
| 8 | Vertex x-coordinate = –b/2a; use to find min/max value quickly. |
| 9 | In option elimination, substitute easy integers (0,1,–1) to strike 2–3 wrong choices fast. |
| 10 | Railway exams rarely test complex roots; focus on real, rational, equal-root cases. |
BETA
