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S3 Maths: Quadratic Equations – Formula and Discriminant (Paper 2 (intermediate))
S3 Maths: Quadratic Equations – Formula and Discriminant (Paper 2 (intermediate))
Paper 2 (intermediate): ten quadratic-equation questions with answers and one-line notes.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 |
| Suggested time | 20 |
| Difficulty mix | Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1 |
| Chapter | Chapter: S3 Maths: Quadratic Equations – Formula and Discriminant |
| Level | S3 |
02 Questions (10 items)
- FoundationSolve 4x^2 – 12x + 9 = 0.
- FoundationSolve x^2 – 8x + 12 = 0.
- FoundationSolve x^2 + x – 12 = 0.
- IntermediateSolve 5x^2 – 3x – 2 = 0.
- IntermediateSolve 2x^2 – 6x + 1 = 0, leaving the answer in surd form.
- IntermediateSolve 3x^2 – 4x + 1 = 0.
- ChallengeDescribe the nature of the roots of 2x^2 – 5x + 4 = 0.
- ChallengeFind k so that x^2 + kx + 9 = 0 has equal roots.
- ChallengeIf alpha and beta are the roots of 2x^2 – 7x + 3 = 0, find alpha + beta and alpha beta.
- Error redoSolving 2x^2 – 5x + 2 = 0, a student writes only (5 + 3) over 4 = 2 and stops. What is missing?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | x = 3/2, a repeated root | The discriminant is 144 minus 144, which is zero, so there is a single root and 12 over 8 simplifies to 3/2. |
| 2 | x = 2 or x = 6 | The discriminant is 64 minus 48, which is 16, so the formula gives (8 plus or minus 4) over 2. |
| 3 | x = 3 or x = -4 | The discriminant is 1 plus 48, which is 49, so the formula gives (-1 plus or minus 7) over 2. |
| 4 | x = 1 or x = -0.4 | The discriminant is 9 plus 40, which is 49, so the formula gives (3 plus or minus 7) over 10. |
| 5 | x = (3 + sqrt(7))/2 or x = (3 – sqrt(7))/2 | The discriminant is 36 minus 8, which is 28; root 28 is 2 root 7, which cancels with the 4 in the denominator. |
| 6 | x = 1 or x = 1/3 | The discriminant is 16 minus 12, which is 4, so the formula gives (4 plus or minus 2) over 6. |
| 7 | No real roots | The discriminant is 25 minus 32, which is -7; negative values mean the curve never meets the x-axis. |
| 8 | k = 6 or k = -6 | Equal roots need the discriminant k^2 – 36 to be zero, so k is plus or minus 6. |
| 9 | alpha + beta = 7/2 and alpha beta = 3/2 | The sum is minus b over a, giving 7/2, and the product is c over a, giving 3/2. |
| 10 | The minus branch: the correct answer is x = 2 or x = 1/2 | The formula has two branches; the square root of the discriminant is 3, and the other branch is (5 minus 3) over 4. |
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.
Before using the formula, rearrange the equation into ax^2 + bx + c = 0 and write down a, b and c clearly, because signs slip most often here. Work out the discriminant D = b^2 – 4ac first: positive means two distinct real roots, zero means a repeated root and negative means no real roots. Remember the plus-or-minus sign gives two branches, simplify the surd before cancelling with the denominator, and finish by substituting the answers back into the original equation.Tutor Registration →
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