Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 73 · 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.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 125 (c) 325 (d) 225
2. 2.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 73 (d) 17
3. 3.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
4. 4.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 9 (c) 24 (d) 10
5. 5.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)
6. 6.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 9216 (c) 10826 (d) 10816
7. 7.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 5625 (c) 4235 (d) 4225
8. 8.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 271 (c) 181 (d) 1
9. 9.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 289 (c) 5 (d) 25
10. 10.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
11. 11.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
12. 12.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
13. 13.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 884 (c) 1156 (d) 686
14. 14.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 4096 (c) 3616 (d) 3136
15. 15.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 225 (d) 657
16. 16.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 2 (c) 4 (d) 202
17. 17.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 181 (d) 271
18. 18.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 36 (c) 64 (d) 6
19. 19.If a + b = 6 and ab = 8, what is (a − b)²? (a) 36 (b) 2 (c) 4 (d) 20
20. 20.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 8 (c) 9 (d) 7
21. 21.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 334 (c) 396 (d) 484
22. 22.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
23. 23.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5635 (c) 4225 (d) 5625
24. 24.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 8649 (c) 8659 (d) 7569
25. 25.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 324 (c) 484 (d) 404
26. 26.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2714 (c) 2704 (d) 2504
27. 27.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
28. 28.If a + b = 14 and ab = 33, what is a² + b²? (a) 130 (b) 163 (c) 196 (d) 262
29. 29.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x − 2)(x − 4)
30. 30.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9614 (c) 9996 (d) 9604
31. 31.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 9)
32. 32.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 181 (c) 61 (d) 121
33. 33.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 100 (c) 91 (d) 82
34. 34.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 529 (c) 289 (d) 391
35. 35.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
36. 36.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 10619 (c) 9409 (d) 10009
37. 37.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 249 (c) 169 (d) 89
38. 38.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 1 and 28 (c) 1 and 10 (d) 2 and 14
39. 39.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 7 and 9 (d) 3 and 21
40. 40.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2035 (c) 2475 (d) 2025

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