Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 327 · 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 = 5 and ab = 6, what is a² + b²? (a) 37 (b) 19 (c) 13 (d) 25
2. 2.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)
3. 3.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 1089 (c) 891 (d) 729
4. 4.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 4 (c) 2 (d) 164
5. 5.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8091 (c) 7569 (d) 8649
6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
7. 7.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 1)(x + 9) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)
8. 8.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 3 (c) 441 (d) 9
9. 9.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x − 2)(x − 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
10. 10.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2714 (c) 2704 (d) 2504
11. 11.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3731 (c) 3721 (d) 3601
12. 12.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 193 (c) 157 (d) 169
13. 13.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10001 (c) 10201 (d) 9801
14. 14.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 71 (d) 73
15. 15.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 14 (c) 12 (d) 13
16. 16.If a + b = 10 and ab = 16, what is (a − b)²? (a) 36 (b) 100 (c) 6 (d) 68
17. 17.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 6 and 9 (c) 1 and 14 (d) 1 and 54
18. 18.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 100 (c) 196 (d) 292
19. 19.What is the sum of the roots of x² − 9x + 14 = 0? (a) 9 (b) 10 (c) 14 (d) 8
20. 20.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 4)(x + 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
21. 21.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 2035 (c) 3025 (d) 2025
22. 22.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1521 (c) 1681 (d) 1599
23. 23.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 11 (c) 30 (d) 10
24. 24.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 635 (c) 875 (d) 625
25. 25.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 361 (c) 399 (d) 441
26. 26.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 484 (c) 324 (d) 404
27. 27.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 1)(x + 24) (d) (x + 3)(x + 8)
28. 28.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 106 (c) 196 (d) 286
29. 29.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 576 (c) 586 (d) 256
30. 30.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 72 (d) 18
31. 31.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6734 (c) 6724 (d) 6404
32. 32.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 12 (c) 13 (d) 27
33. 33.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 225 (c) 441 (d) 333
34. 34.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 145 (d) 289
35. 35.What are the roots of x² − 9x + 20 = 0? (a) 4 and 5 (b) 2 and 10 (c) 1 and 8 (d) 1 and 20
36. 36.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
37. 37.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 144 (c) 198 (d) 90
38. 38.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
39. 39.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
40. 40.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)

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