Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 49 · 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 = 28, what is (a − b)²? (a) 9 (b) 3 (c) 65 (d) 121
2. 2.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2516 (c) 2926 (d) 2116
3. 3.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 193 (c) 130 (d) 382
4. 4.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4106 (c) 3616 (d) 4096
5. 5.What are the roots of x² − 12x + 27 = 0? (a) 1 and 11 (b) 2 and 10 (c) 1 and 27 (d) 3 and 9
6. 6.What is the product of the roots of x² − 14x + 45 = 0? (a) 45 (b) 14 (c) 46 (d) 44
7. 7.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)
8. 8.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
9. 9.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 256 (c) 576 (d) 416
10. 10.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 7225 (c) 5625 (d) 5635
11. 11.What is the product of the roots of x² − 9x + 20 = 0? (a) 20 (b) 21 (c) 19 (d) 9
12. 12.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 8 (c) 9 (d) 14
13. 13.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 1 (d) 361
14. 14.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
15. 15.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 208 (c) 400 (d) 304
16. 16.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 1)(x + 4) (d) (x + 2)(x + 3)
17. 17.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
18. 18.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5486 (c) 4916 (d) 5476
19. 19.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 117 (c) 198 (d) 144
20. 20.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 289 (c) 539 (d) 409
21. 21.If a + b = 13 and ab = 22, what is a² + b²? (a) 147 (b) 213 (c) 169 (d) 125
22. 22.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 256 (c) 4 (d) 16
23. 23.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
24. 24.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 1 and 56 (d) 7 and 8
25. 25.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 1024 (c) 784 (d) 896
26. 26.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
27. 27.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 1 (c) 13 (d) 19
28. 28.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 5)(x + 9)
29. 29.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 181 (c) 61 (d) 121
30. 30.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 13 (c) 40 (d) 14
31. 31.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 3 and 5 (c) 1 and 7 (d) 2 and 6
32. 32.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 441 (c) 361 (d) 399
33. 33.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 433 (c) 289 (d) 145
34. 34.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 225 (c) 189 (d) 153
35. 35.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
36. 36.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 25 (c) 37 (d) 73
37. 37.If a + b = 8 and ab = 12, what is a² + b²? (a) 40 (b) 88 (c) 64 (d) 52
38. 38.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 1 and 16 (d) 2 and 36
39. 39.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6401 (c) 6571 (d) 6241
40. 40.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 10816 (c) 10016 (d) 9216

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