Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 980 · 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 = 30, what is (a − b)²? (a) 121 (b) 61 (c) 1 (d) 91
2. 2.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4634 (c) 4896 (d) 4624
3. 3.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 40)
4. 4.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
5. 5.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 36 (c) 144 (d) 6
6. 6.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 19 (c) 13 (d) 37
7. 7.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 121 (c) 111 (d) 101
8. 8.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 125 (c) 225 (d) 325
9. 9.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3025 (c) 2525 (d) 3035
10. 10.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 1 (d) 181
11. 11.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 2025 (c) 1225 (d) 1235
12. 12.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8091 (c) 8649 (d) 7569
13. 13.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 12 (c) 35 (d) 11
14. 14.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 15 (d) 57
15. 15.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 262 (c) 196 (d) 130
16. 16.If a + b = 8 and ab = 12, what is (a − b)²? (a) 40 (b) 4 (c) 64 (d) 16
17. 17.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 63 (c) 117 (d) 45
18. 18.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 539 (c) 409 (d) 289
19. 19.If a + b = 9 and ab = 8, what is (a − b)²? (a) 7 (b) 81 (c) 49 (d) 65
20. 20.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
21. 21.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 896 (c) 784 (d) 1024
22. 22.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
23. 23.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
24. 24.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x + 1)(x + 14)
25. 25.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 169 (c) 85 (d) 253
26. 26.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 6 (c) 36 (d) 324
27. 27.If a + b = 12 and ab = 11, what is a² + b²? (a) 122 (b) 144 (c) 133 (d) 166
28. 28.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 32 (c) 4 and 8 (d) 1 and 11
29. 29.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 20 (c) 9 (d) 21
30. 30.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 35 (c) 36 (d) 13
31. 31.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 8649 (c) 7569 (d) 8109
32. 32.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
33. 33.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 71 (d) 17
34. 34.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2401 (c) 2411 (d) 2601
35. 35.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 14 (c) 36 (d) 13
36. 36.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 19 (c) 18 (d) 17
37. 37.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
38. 38.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 7 and 9 (c) 3 and 21 (d) 1 and 63
39. 39.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 196 (c) 10 (d) 100
40. 40.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 5476 (c) 5486 (d) 4356

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