Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 363 · 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 sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 7 (c) 6 (d) 5
2. 2.What are the roots of x² − 12x + 27 = 0? (a) 1 and 11 (b) 3 and 9 (c) 2 and 10 (d) 1 and 27
3. 3.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x − 4)(x − 7)
4. 4.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 1 and 10 (c) 2 and 14 (d) 4 and 7
5. 5.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 97 (c) 133 (d) 241
6. 6.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 5 (c) 25 (d) 121
7. 7.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 151 (c) 286 (d) 196
8. 8.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 484 (c) 324 (d) 334
9. 9.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 24 (c) 23 (d) 10
10. 10.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 592 (c) 400 (d) 208
11. 11.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
12. 12.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
13. 13.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 9 (c) 7 (d) 8
14. 14.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 56 (d) 57
15. 15.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8464 (c) 7744 (d) 8104
16. 16.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 1)(x + 32) (d) (x + 4)(x + 8)
17. 17.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 1 and 72 (d) 8 and 9
18. 18.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8101 (c) 8291 (d) 7921
19. 19.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 17 (d) 73
20. 20.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 1 and 4 (c) 1 and 6 (d) 2 and 4
21. 21.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 1 and 56 (d) 7 and 8
22. 22.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 8 (c) 6 (d) 7
23. 23.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 14 (c) 12 (d) 13
24. 24.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 185 (d) 273
25. 25.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)
26. 26.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 971 (c) 961 (d) 841
27. 27.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 598 (c) 202 (d) 400
28. 28.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 484 (d) 324
29. 29.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 225 (c) 153 (d) 81
30. 30.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 289 (c) 409 (d) 539
31. 31.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 2 (c) 20 (d) 36
32. 32.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
33. 33.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
34. 34.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 219 (c) 149 (d) 429
35. 35.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4366 (c) 5476 (d) 4884
36. 36.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1616 (c) 1936 (d) 1296
37. 37.If a + b = 11 and ab = 24, what is a² + b²? (a) 73 (b) 121 (c) 169 (d) 97
38. 38.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 193 (c) 145 (d) 169
39. 39.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 16 (c) 256 (d) 4
40. 40.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3146 (c) 3136 (d) 3584

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