Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 724 · 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 + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
2. 2.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)
3. 3.If a + b = 7 and ab = 10, what is a² + b²? (a) 39 (b) 49 (c) 69 (d) 29
4. 4.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 13 (c) 12 (d) 11
5. 5.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 18 (c) 17 (d) 19
6. 6.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 5)(x + 7) (c) (x − 5)(x − 7) (d) (x + 1)(x + 11)
7. 7.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 2 and 36 (d) 8 and 9
8. 8.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 324 (c) 334 (d) 484
9. 9.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 2 and 24 (c) 6 and 8 (d) 1 and 48
10. 10.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 2 (c) 144 (d) 4
11. 11.If a + b = 13 and ab = 30, what is (a − b)²? (a) 169 (b) 49 (c) 7 (d) 109
12. 12.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6399 (c) 6241 (d) 6251
13. 13.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 6899 (c) 5929 (d) 6409
14. 14.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 49 (d) 47
15. 15.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 32 (c) 13 (d) 12
16. 16.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 151 (c) 286 (d) 196
17. 17.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 451 (d) 441
18. 18.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 97 (c) 73 (d) 169
19. 19.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 841 (c) 901 (d) 971
20. 20.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 1 and 28 (c) 1 and 10 (d) 2 and 14
21. 21.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 371 (d) 361
22. 22.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
23. 23.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4896 (c) 5184 (d) 4624
24. 24.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
25. 25.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 9 (c) 361 (d) 185
26. 26.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 15 (d) 14
27. 27.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3025 (c) 2525 (d) 3035
28. 28.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 5184 (c) 4904 (d) 5194
29. 29.If a + b = 9 and ab = 8, what is a² + b²? (a) 65 (b) 73 (c) 81 (d) 97
30. 30.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 72 (d) 73
31. 31.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 117 (c) 144 (d) 198
32. 32.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 121 (c) 25 (d) 5
33. 33.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
34. 34.If a + b = 12 and ab = 20, what is (a − b)²? (a) 104 (b) 64 (c) 8 (d) 144
35. 35.If a + b = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 11 (c) 169 (d) 121
36. 36.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
37. 37.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
38. 38.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
39. 39.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 39 (c) 41 (d) 40
40. 40.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7225 (b) 6425 (c) 7235 (d) 5625

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