Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 341 · 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.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1099 (c) 729 (d) 1089
2. 2.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 10)
3. 3.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 13 (c) 15 (d) 14
4. 4.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 88 (c) 52 (d) 40
5. 5.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
6. 6.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 15 (c) 48 (d) 14
7. 7.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 62 (d) 64
8. 8.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
9. 9.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1444 (c) 1596 (d) 1764
10. 10.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 32 (c) 11 (d) 12
11. 11.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 361 (c) 273 (d) 185
12. 12.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 9 (c) 18 (d) 8
13. 13.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 76 (c) 148 (d) 52
14. 14.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 1 and 10 (c) 4 and 7 (d) 2 and 14
15. 15.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 2 (c) 144 (d) 4
16. 16.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 286 (c) 106 (d) 196
17. 17.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 169 (c) 225 (d) 1
18. 18.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 13 (c) 42 (d) 12
19. 19.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
20. 20.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 181 (d) 271
21. 21.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 14 (c) 15 (d) 13
22. 22.What are the roots of x² − 14x + 45 = 0? (a) 5 and 9 (b) 1 and 45 (c) 1 and 13 (d) 3 and 15
23. 23.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1681 (c) 1521 (d) 1599
24. 24.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 4901 (c) 5041 (d) 5051
25. 25.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 451 (d) 441
26. 26.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 539 (c) 409 (d) 529
27. 27.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)
28. 28.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 81 (c) 61 (d) 1
29. 29.If a + b = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 324 (c) 6 (d) 180
30. 30.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 16 (c) 4 (d) 196
31. 31.If a + b = 11 and ab = 28, what is (a − b)²? (a) 121 (b) 9 (c) 65 (d) 3
32. 32.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 81 (c) 117 (d) 63
33. 33.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 6 (c) 146 (d) 36
34. 34.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 2 and 12 (c) 1 and 9 (d) 4 and 6
35. 35.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2611 (c) 2601 (d) 2401
36. 36.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 18 (d) 16
37. 37.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 6 and 8 (c) 1 and 48 (d) 1 and 13
38. 38.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
39. 39.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3584 (c) 3146 (d) 4096
40. 40.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 145 (c) 193 (d) 157

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