Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 450 · 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.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
2. 2.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 6391 (c) 5929 (d) 5939
3. 3.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2219 (c) 2209 (d) 2809
4. 4.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 4 (c) 324 (d) 16
5. 5.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 478 (c) 324 (d) 247
6. 6.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1601 (c) 1691 (d) 1681
7. 7.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 4 and 9 (c) 2 and 18 (d) 1 and 12
8. 8.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 18 (d) 17
9. 9.What is the product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 8 (c) 9 (d) 6
10. 10.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 12)
11. 11.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 1)(x + 24)
12. 12.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 13 (d) 14
13. 13.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
14. 14.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 16 (d) 14
15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
16. 16.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 8 and 9 (d) 2 and 36
17. 17.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 12 (c) 14 (d) 40
18. 18.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
19. 19.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 904 (c) 1034 (d) 784
20. 20.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 5476 (c) 4356 (d) 4366
21. 21.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
22. 22.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 29 (c) 69 (d) 39
23. 23.If a + b = 8 and ab = 12, what is (a − b)²? (a) 40 (b) 64 (c) 4 (d) 16
24. 24.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 1)(x + 7)
25. 25.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2714 (c) 2504 (d) 2704
26. 26.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 25 (c) 5 (d) 193
27. 27.What are the roots of x² − 7x + 10 = 0? (a) 1 and 6 (b) 3 and 4 (c) 1 and 10 (d) 2 and 5
28. 28.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 12 (c) 27 (d) 11
29. 29.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 13 (c) 14 (d) 12
30. 30.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 214 (c) 74 (d) 109
31. 31.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 181 (c) 121 (d) 61
32. 32.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 142 (c) 100 (d) 79
33. 33.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 130 (c) 163 (d) 262
34. 34.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)
35. 35.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 7569 (c) 8109 (d) 8649
36. 36.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
37. 37.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 172 (c) 148 (d) 196
38. 38.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 7579 (c) 8649 (d) 8091
39. 39.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 117 (c) 3 (d) 225
40. 40.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 1)(x + 16) (d) (x + 2)(x + 8)

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