Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 400 · 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 = 17 and ab = 72, what is a² + b²? (a) 145 (b) 289 (c) 433 (d) 217
2. 2.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 15 (c) 53 (d) 54
3. 3.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 9226 (c) 10816 (d) 9984
4. 4.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
5. 5.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 4 (d) 2
6. 6.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 13 (c) 48 (d) 15
7. 7.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
8. 8.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 121 (c) 9 (d) 65
9. 9.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 884 (c) 1156 (d) 676
10. 10.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)
11. 11.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 19 (c) 25 (d) 1
12. 12.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1609 (c) 1859 (d) 1369
13. 13.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4096 (c) 4106 (d) 3616
14. 14.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1296 (c) 1946 (d) 1936
15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
16. 16.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 100 (c) 52 (d) 148
17. 17.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)
18. 18.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 13 (c) 22 (d) 10
19. 19.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 1)(x + 11)
20. 20.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 9025 (c) 7225 (d) 9035
21. 21.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 25 (c) 73 (d) 49
22. 22.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 4 (c) 170 (d) 324
23. 23.What are the roots of x² − 14x + 45 = 0? (a) 5 and 9 (b) 3 and 15 (c) 1 and 13 (d) 1 and 45
24. 24.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 5329 (c) 4909 (d) 4489
25. 25.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3601 (c) 3731 (d) 3721
26. 26.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 625 (c) 235 (d) 225
27. 27.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 5786 (c) 6384 (d) 7056
28. 28.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 784 (c) 1034 (d) 1024
29. 29.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x + 1)(x + 6)
30. 30.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 9 (c) 10 (d) 11
31. 31.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 1)(x + 10)
32. 32.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 49 (c) 69 (d) 39
33. 33.If a + b = 11 and ab = 18, what is a² + b²? (a) 85 (b) 103 (c) 121 (d) 157
34. 34.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
35. 35.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6724 (c) 6404 (d) 6084
36. 36.If a + b = 16 and ab = 60, what is a² + b²? (a) 196 (b) 256 (c) 376 (d) 136
37. 37.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)
38. 38.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4356 (c) 4884 (d) 5476
39. 39.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 3 (c) 89 (d) 9
40. 40.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 244 (d) 164

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