Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 662 · 40 questions

How to solve it (Use the identity, skip the guessing): (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; so a² + b² = (a + b)² − 2ab and (a − b)² = (a + b)² − 4ab. For x² − Sx + P = 0: the sum of the roots is S and the product is P. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

1. 1.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 16 (c) 14 (d) 15
2. 2.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 7 (c) 6 (d) 12
3. 3.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 25 (c) 9 (d) 17
4. 4.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 1 and 16 (d) 2 and 36
5. 5.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 1 and 56 (c) 2 and 28 (d) 7 and 8
6. 6.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x − 4)(x − 7)
7. 7.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 6 (c) 10 (d) 7
8. 8.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 1 and 13 (c) 2 and 24 (d) 6 and 8
9. 9.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 2 (c) 202 (d) 400
10. 10.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2501 (c) 2601 (d) 2401
11. 11.What is the sum of the roots of x² − 14x + 45 = 0? (a) 14 (b) 15 (c) 45 (d) 13
12. 12.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 35 (c) 36 (d) 13
13. 13.If a + b = 12 and ab = 32, what is (a − b)²? (a) 4 (b) 144 (c) 80 (d) 16
14. 14.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x + 1)(x + 14)
15. 15.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
16. 16.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
17. 17.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6571 (c) 6561 (d) 6241
18. 18.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1849 (c) 1591 (d) 1369
19. 19.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 21 (c) 33 (d) 25
20. 20.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
21. 21.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 16 (c) 40 (d) 4
22. 22.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 9 (c) 8 (d) 15
23. 23.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 10 (c) 12 (d) 11
24. 24.If a + b = 6 and ab = 8, what is (a − b)²? (a) 36 (b) 20 (c) 2 (d) 4
25. 25.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 1 and 11 (c) 2 and 10 (d) 5 and 7
26. 26.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 7) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
27. 27.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 1 (d) 181
28. 28.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 9 (c) 1 (d) 7
29. 29.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 3 (c) 9 (d) 45
30. 30.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
31. 31.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 103 (c) 85 (d) 157
32. 32.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
33. 33.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 2 (c) 4 (d) 74
34. 34.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 2)(x + 9) (c) (x + 1)(x + 18) (d) (x − 2)(x − 9)
35. 35.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 45) (c) (x + 1)(x + 13) (d) (x − 5)(x − 9)
36. 36.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 451 (c) 361 (d) 441
37. 37.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 324 (c) 252 (d) 468
38. 38.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3025 (c) 2525 (d) 3035
39. 39.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 23 (c) 11 (d) 24
40. 40.What is the product of the roots of x² − 12x + 32 = 0? (a) 33 (b) 32 (c) 12 (d) 31

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