Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 732 · 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 + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 1)(x + 7)
2. 2.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8281 (c) 8291 (d) 8101
3. 3.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 1 and 9 (c) 3 and 7 (d) 2 and 8
4. 4.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 5 (c) 193 (d) 25
5. 5.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 3 (c) 121 (d) 9
6. 6.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 1 and 56 (d) 7 and 8
7. 7.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1225 (c) 1625 (d) 2025
8. 8.If a + b = 12 and ab = 27, what is (a − b)²? (a) 6 (b) 90 (c) 36 (d) 144
9. 9.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 202 (c) 598 (d) 301
10. 10.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 539 (c) 529 (d) 409
11. 11.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1936 (c) 1616 (d) 1296
12. 12.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 8 (c) 15 (d) 14
13. 13.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 42 (d) 1 and 12
14. 14.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 14 (c) 12 (d) 42
15. 15.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 1099 (c) 909 (d) 729
16. 16.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6399 (c) 6561 (d) 6241
17. 17.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 28)
18. 18.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 1 and 13 (c) 2 and 24 (d) 6 and 8
19. 19.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2475 (c) 2025 (d) 2035
20. 20.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 729 (c) 891 (d) 1089
21. 21.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 9 (c) 17 (d) 25
22. 22.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 64 (c) 4 (d) 2
23. 23.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 56 (d) 15
24. 24.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 61 (c) 37 (d) 49
25. 25.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
26. 26.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
27. 27.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 478 (c) 170 (d) 247
28. 28.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 45) (c) (x + 1)(x + 13) (d) (x + 5)(x + 9)
29. 29.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 541 (c) 361 (d) 271
30. 30.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 9 (c) 7 (d) 15
31. 31.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 45 (c) 44 (d) 46
32. 32.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 2 and 10 (c) 3 and 9 (d) 1 and 11
33. 33.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
34. 34.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 361 (d) 271
35. 35.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 36 (c) 20 (d) 28
36. 36.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 184 (c) 104 (d) 124
37. 37.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 361 (c) 441 (d) 399
38. 38.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 2 and 20 (c) 1 and 40 (d) 5 and 8
39. 39.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 144 (c) 117 (d) 90
40. 40.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 225 (c) 169 (d) 113

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