A quick practice test on Quadratic Equations to check how ready you are. Attempt it fully on paper first — give yourself about 45 minutes — then reveal the answer key at the bottom to mark yourself. Total: 20 marks. Need a refresher first? Read the Quadratic Equations notes.
Section A — Objective (5 × 1 mark)
- The standard form of a quadratic equation is: (a) ax + b = 0 (b) ax² + bx + c = 0, a ≠ 0 (c) ax³ + bx² + c = 0 (d) x + c = 0
- A quadratic equation can have at most how many roots? (a) 1 (b) 2 (c) 3 (d) 4
- The discriminant of ax² + bx + c = 0 is: (a) b² − 4ac (b) b² + 4ac (c) 4ac − b² (d) 2a
- If the discriminant is negative, the roots are: (a) real and equal (b) real and distinct (c) not real (d) zero
- The roots of x² − 9 = 0 are: (a) 3, −3 (b) 9, −9 (c) 3, 3 (d) 9 only
Section B — Short Answer (5 × 2 marks)
- Solve by factorisation: x² − 7x + 12 = 0.
- Find the discriminant of 2x² − 3x + 1 = 0 and state the nature of its roots.
- Find the value of k for which x² − 6x + k = 0 has equal roots.
- Solve: x² − 5x + 6 = 0.
- Does x² + 4 = 0 have real roots? Give a reason.
Section C — Long Answer (1 × 5 marks)
- The sum of a number and its reciprocal is 10/3. Find the number.
Show the Full Answer Key
Section A: 1 → (b) 2 → (b) 3 → (a) 4 → (c) 5 → (a).
6. x² − 7x + 12 = (x − 3)(x − 4) = 0, so x = 3 or x = 4.
7. D = (−3)² − 4(2)(1) = 9 − 8 = 1, which is greater than 0, so the equation has two distinct real roots.
8. Equal roots means D = 0: 36 − 4k = 0 → k = 9.
9. x² − 5x + 6 = (x − 2)(x − 3) = 0, so x = 2 or x = 3.
10. No. Here D = 0 − 16 = −16, which is less than 0, so there are no real roots.
11. Let the number be x. Then x + 1/x = 10/3. Multiplying through by 3x: 3x² + 3 = 10x → 3x² − 10x + 3 = 0 → (3x − 1)(x − 3) = 0 → x = 1/3 or x = 3. So the number is 3 (or 1/3).
How many did you get? Whatever the score, note the ones you missed and try them again tomorrow — one more correct than today is the whole game.