Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 841 · 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 + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 27) (d) (x + 1)(x + 11)
2. 2.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
3. 3.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 896 (d) 1024
4. 4.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 2)(x + 5)
5. 5.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4884 (c) 4356 (d) 4366
6. 6.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
7. 7.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 15 (c) 14 (d) 54
8. 8.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 3 and 4 (c) 2 and 6 (d) 1 and 7
9. 9.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 10 (c) 12 (d) 30
10. 10.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 334 (c) 484 (d) 396
11. 11.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 8116 (c) 7396 (d) 8846
12. 12.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6401 (c) 6241 (d) 6561
13. 13.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 5 (c) 73 (d) 121
14. 14.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)
15. 15.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7931 (c) 7921 (d) 8099
16. 16.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
17. 17.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8104 (c) 8464 (d) 7744
18. 18.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 7 and 8 (d) 1 and 14
19. 19.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 72 (c) 1 and 16 (d) 2 and 36
20. 20.If a + b = 13 and ab = 36, what is (a − b)²? (a) 169 (b) 25 (c) 97 (d) 5
21. 21.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 148 (c) 52 (d) 76
22. 22.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 625 (c) 225 (d) 235
23. 23.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 1 and 9 (c) 2 and 8 (d) 1 and 21
24. 24.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
25. 25.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4624 (c) 4634 (d) 4896
26. 26.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 729 (c) 1089 (d) 909
27. 27.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 7) (c) (x − 2)(x − 6) (d) (x + 1)(x + 12)
28. 28.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 36 (c) 34 (d) 35
29. 29.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 81 (c) 117 (d) 63
30. 30.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 74 (c) 2 (d) 4
31. 31.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 145 (c) 433 (d) 217
32. 32.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 47 (c) 14 (d) 48
33. 33.If a + b = 11 and ab = 18, what is (a − b)²? (a) 85 (b) 121 (c) 49 (d) 7
34. 34.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 225 (c) 125 (d) 175
35. 35.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
36. 36.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
37. 37.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 12 (c) 11 (d) 18
38. 38.If a + b = 16 and ab = 48, what is a² + b²? (a) 208 (b) 352 (c) 160 (d) 256
39. 39.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 52 (c) 64 (d) 40
40. 40.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 301 (c) 400 (d) 598

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