中文（繁體）
中文（简体）
English
S4 Quadratic Equations, Paper 3: discriminants and word problems
S4 Quadratic Equations, Paper 3: discriminants and word problems
The discriminant settles how many real roots exist, while word problems ask you to build the equation first.
Focus Judge the roots, solve the equation, and turn each word problem into a quadratic before solving.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 |
| Suggested time | 25 |
| Difficulty mix | Foundation 4 · Intermediate 3 · Challenge 2 · Error redo 1 |
| Chapter | Chapter: S4 Maths: Quadratic Equations |
| Level | S4 |
02 Questions (10 items)
- FoundationSolve x² − 7x + 12 = 0.
- FoundationSolve x² + 5x = 0.
- FoundationSolve 2x² = 18.
- FoundationSolve (x − 5)² = 4.
- IntermediateWithout solving, decide whether x² + 4x + 7 = 0 has real roots.
- IntermediateSolve x² − 6x + 4 = 0 with the formula, leaving surds.
- IntermediateIf x² + kx + 9 = 0 has equal roots and k > 0, find k.
- Challengeα and β are the roots of x² − 5x + 3 = 0. Find α² + β² and 1/α + 1/β.
- ChallengeA rectangle is 3 cm longer than it is wide and has area 40 cm². Find the width.
- Error redoError: solving x² = 3x, a student divides by x and answers x = 3. Which root is missing?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | x = 3 or x = 4 | Factorising gives (x minus 3)(x minus 4), so the roots are 3 and 4. |
| 2 | x = 0 or x = −5 | Taking out x gives x(x plus 5), so the roots are zero and minus 5. |
| 3 | x = 3 or x = −3 | Divide by 2 to get x squared equals 9, then write both signs. |
| 4 | x = 7 or x = 3 | The square root gives x minus 5 equals plus or minus 2, so 7 or 3. |
| 5 | No real roots, discriminant −12 | The discriminant is 16 minus 28, which is minus 12, so there are no real roots. |
| 6 | x = 3 ± √5 | The discriminant is 36 minus 16, giving 20, and its root is 2 root 5. |
| 7 | k = 6 | Equal roots need k squared minus 36 to be zero, so k is plus or minus 6 and we take 6. |
| 8 | 19 and 5/3 | The sum is 5 and the product 3, so α² plus β² is 19 and the reciprocal sum is 5 over 3. |
| 9 | Width 5 cm, length 8 cm | Let the width be x, so x(x plus 3) is 40, giving (x plus 8)(x minus 5) and width 5. |
| 10 | The root x = 0 | Dividing by x before ruling out zero loses a root; move everything over to get x(x minus 3). |
04 What to do after marking
- After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
- Repeat the same item two days later; move to the next paper only when both attempts are correct.
- Parents only need the answer and note columns; there is no need to explain every item.
Book an S4 Maths tutorTutor Registration →
Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.
