Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 277 · 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 + 10x + 21? (a) (x + 1)(x + 21) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 9)
2. 2.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 3136 (c) 4096 (d) 3146
3. 3.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)
4. 4.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6401 (c) 6571 (d) 6241
5. 5.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 352 (c) 160 (d) 208
6. 6.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5476 (c) 5486 (d) 4916
7. 7.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
8. 8.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 3 and 21 (d) 7 and 9
9. 9.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 172 (c) 148 (d) 196
10. 10.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
11. 11.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 1 and 16 (c) 3 and 7 (d) 2 and 8
12. 12.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 85 (c) 253 (d) 127
13. 13.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)
14. 14.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 2475 (c) 3025 (d) 2025
15. 15.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 576 (d) 586
16. 16.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 157 (c) 289 (d) 25
17. 17.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
18. 18.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8109 (c) 7569 (d) 8659
19. 19.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 9 (c) 10 (d) 21
20. 20.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 63 (d) 17
21. 21.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1575 (c) 1235 (d) 1225
22. 22.If a + b = 13 and ab = 36, what is (a − b)²? (a) 25 (b) 5 (c) 97 (d) 169
23. 23.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 2 and 8 (c) 1 and 21 (d) 3 and 7
24. 24.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 4 (c) 208 (d) 16
25. 25.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 2 and 18 (c) 1 and 12 (d) 4 and 9
26. 26.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 961 (d) 901
27. 27.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
28. 28.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 17 (d) 73
29. 29.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 198 (c) 144 (d) 117
30. 30.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1099 (c) 1089 (d) 729
31. 31.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 441 (c) 333 (d) 225
32. 32.What is the product of the roots of x² − 13x + 40 = 0? (a) 39 (b) 41 (c) 13 (d) 40
33. 33.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 117 (c) 225 (d) 3
34. 34.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)
35. 35.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
36. 36.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3481 (c) 3601 (d) 3731
37. 37.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 181 (c) 61 (d) 121
38. 38.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 9 (c) 225 (d) 441
39. 39.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)
40. 40.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 2 and 28 (c) 1 and 14 (d) 1 and 56

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