Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 750 · 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 are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 8 and 9 (d) 1 and 72
2. 2.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 74 (c) 144 (d) 109
3. 3.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 289 (c) 217 (d) 433
4. 4.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
5. 5.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 9 (d) 10
6. 6.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 53 (c) 54 (d) 55
7. 7.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 2)(x + 3) (c) (x + 1)(x + 6) (d) (x − 2)(x − 3)
8. 8.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
9. 9.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
10. 10.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1599 (c) 1521 (d) 1531
11. 11.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 10001 (d) 9801
12. 12.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 2 and 12 (c) 3 and 8 (d) 1 and 24
13. 13.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1369 (c) 1849 (d) 1379
14. 14.If a + b = 8 and ab = 7, what is a² + b²? (a) 78 (b) 57 (c) 64 (d) 50
15. 15.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x − 4)(x − 8)
16. 16.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 53 (c) 67 (d) 109
17. 17.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4891 (b) 4499 (c) 5329 (d) 4489
18. 18.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 337 (d) 225
19. 19.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 48 (c) 14 (d) 15
20. 20.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 2)(x + 6) (c) (x − 2)(x − 6) (d) (x + 1)(x + 12)
21. 21.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 529 (c) 193 (d) 277
22. 22.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x − 2)(x − 7) (c) (x + 1)(x + 14) (d) (x + 2)(x + 7)
23. 23.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 537 (c) 273 (d) 361
24. 24.If a + b = 15 and ab = 36, what is a² + b²? (a) 189 (b) 153 (c) 297 (d) 225
25. 25.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4366 (c) 4884 (d) 5476
26. 26.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 366 (c) 256 (d) 146
27. 27.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 9025 (c) 11035 (d) 11025
28. 28.If a + b = 11 and ab = 18, what is a² + b²? (a) 85 (b) 121 (c) 157 (d) 103
29. 29.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 225 (c) 441 (d) 657
30. 30.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
31. 31.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 7 and 8 (d) 1 and 56
32. 32.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 144 (c) 122 (d) 133
33. 33.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
34. 34.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 145 (c) 289 (d) 217
35. 35.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
36. 36.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
37. 37.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
38. 38.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x − 4)(x − 9)
39. 39.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3374 (c) 3844 (d) 3364
40. 40.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 625 (c) 235 (d) 375

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