Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 66 · 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 = 11 and ab = 18, what is a² + b²? (a) 85 (b) 103 (c) 157 (d) 121
2. 2.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 539 (c) 289 (d) 529
3. 3.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1024 (c) 1034 (d) 784
4. 4.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 16 (c) 58 (d) 4
5. 5.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 289 (c) 145 (d) 217
6. 6.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 2 and 36 (d) 1 and 72
7. 7.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 7 (c) 10 (d) 11
8. 8.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6571 (c) 6241 (d) 6401
9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
10. 10.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1691 (c) 1521 (d) 1681
11. 11.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1521 (c) 1531 (d) 1599
12. 12.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 8104 (c) 7744 (d) 8474
13. 13.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 181 (c) 61 (d) 91
14. 14.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 9) (c) (x + 1)(x + 21) (d) (x − 3)(x − 7)
15. 15.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 1 (c) 225 (d) 113
16. 16.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 136 (c) 196 (d) 256
17. 17.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
18. 18.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
19. 19.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 26 (c) 46 (d) 36
20. 20.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4356 (c) 4884 (d) 4366
21. 21.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4225 (c) 4875 (d) 5625
22. 22.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 144 (c) 2 (d) 4
23. 23.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 9 (c) 18 (d) 10
24. 24.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x − 2)(x − 3)
25. 25.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 14 (d) 12
26. 26.What are the roots of x² − 13x + 36 = 0? (a) 4 and 9 (b) 2 and 18 (c) 1 and 12 (d) 1 and 36
27. 27.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)
28. 28.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 13 (c) 15 (d) 14
29. 29.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 111 (c) 101 (d) 121
30. 30.What is the product of the roots of x² − 8x + 15 = 0? (a) 15 (b) 8 (c) 16 (d) 14
31. 31.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 127 (c) 1 (d) 169
32. 32.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)
33. 33.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 16 (c) 54 (d) 15
34. 34.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 64 (c) 49 (d) 94
35. 35.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3249 (c) 3969 (d) 3259
36. 36.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 62 (d) 63
37. 37.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 3136 (c) 4096 (d) 4106
38. 38.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 17 (c) 9 (d) 3
39. 39.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 1 and 12 (c) 5 and 8 (d) 1 and 40
40. 40.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 35 (c) 37 (d) 36

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