Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 690 · 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 + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
2. 2.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 4 (c) 6 (d) 7
3. 3.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)
4. 4.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)
5. 5.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1681 (c) 1691 (d) 1601
6. 6.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 16 (d) 15
7. 7.If a + b = 16 and ab = 55, what is (a − b)²? (a) 36 (b) 146 (c) 256 (d) 6
8. 8.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 14 (d) 13
9. 9.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 909 (c) 1099 (d) 729
10. 10.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 13 (c) 6 and 8 (d) 1 and 48
11. 11.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 10816 (c) 9226 (d) 9984
12. 12.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 24) (c) (x + 3)(x + 8) (d) (x + 1)(x + 10)
13. 13.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 8099 (c) 7931 (d) 8281
14. 14.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2304 (c) 2496 (d) 2314
15. 15.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
16. 16.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 11 (d) 13
17. 17.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 94 (c) 34 (d) 49
18. 18.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 262 (c) 130 (d) 163
19. 19.What is the product of the roots of x² − 10x + 16 = 0? (a) 17 (b) 15 (c) 10 (d) 16
20. 20.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 133 (c) 144 (d) 122
21. 21.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
22. 22.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 625 (d) 235
23. 23.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
24. 24.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 5476 (c) 4356 (d) 4916
25. 25.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 225 (c) 441 (d) 333
26. 26.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 1156 (c) 884 (d) 676
27. 27.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 9025 (c) 10025 (d) 11035
28. 28.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 23 (c) 24 (d) 25
29. 29.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 27 (c) 11 (d) 28
30. 30.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 148 (c) 196 (d) 100
31. 31.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2126 (c) 2484 (d) 2116
32. 32.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 56 (c) 7 and 8 (d) 1 and 14
33. 33.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 6425 (c) 7225 (d) 5625
34. 34.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 116 (c) 36 (d) 6
35. 35.If a + b = 6 and ab = 8, what is (a − b)²? (a) 36 (b) 2 (c) 20 (d) 4
36. 36.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4106 (c) 3136 (d) 4096
37. 37.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x − 3)(x − 5) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
38. 38.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 14 (c) 44 (d) 45
39. 39.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 1 (c) 7 (d) 5
40. 40.If a + b = 9 and ab = 8, what is (a − b)²? (a) 65 (b) 7 (c) 81 (d) 49

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