Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 103 · 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 = 12 and ab = 11, what is a² + b²? (a) 133 (b) 122 (c) 166 (d) 144
2. 2.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)
3. 3.If a + b = 16 and ab = 48, what is (a − b)²? (a) 160 (b) 64 (c) 256 (d) 8
4. 4.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2501 (c) 2611 (d) 2401
5. 5.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 49 (c) 7 (d) 85
6. 6.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)
7. 7.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 16 (c) 58 (d) 4
8. 8.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 324 (c) 164 (d) 4
9. 9.If a + b = 13 and ab = 36, what is (a − b)²? (a) 169 (b) 5 (c) 25 (d) 97
10. 10.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 3025 (c) 3625 (d) 4225
11. 11.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
12. 12.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4771 (c) 4899 (d) 4761
13. 13.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 21 (c) 11 (d) 10
14. 14.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 784 (c) 1034 (d) 1024
15. 15.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
16. 16.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 7 (c) 4 (d) 5
17. 17.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 16 (c) 11 (d) 10
18. 18.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3025 (d) 3035
19. 19.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9999 (c) 9811 (d) 10201
20. 20.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 625 (c) 875 (d) 1225
21. 21.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
22. 22.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 5 and 9 (c) 3 and 15 (d) 1 and 13
23. 23.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4096 (c) 4106 (d) 3616
24. 24.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 1 (d) 181
25. 25.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9409 (c) 9419 (d) 9991
26. 26.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 9 (c) 45 (d) 81
27. 27.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 31 (d) 29
28. 28.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 164 (c) 324 (d) 484
29. 29.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4499 (c) 4891 (d) 4489
30. 30.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 185 (c) 273 (d) 361
31. 31.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 4225 (c) 5635 (d) 4925
32. 32.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1235 (c) 1225 (d) 625
33. 33.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 15 (c) 14 (d) 8
34. 34.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
35. 35.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
36. 36.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 26 (c) 4 (d) 16
37. 37.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
38. 38.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 74 (c) 2 (d) 144
39. 39.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 4)(x + 8) (c) (x + 1)(x + 32) (d) (x + 1)(x + 11)
40. 40.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 12 (d) 10

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