Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 988 · 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 75²? (a) 5635 (b) 4225 (c) 4925 (d) 5625
2. 2.What are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 4 and 9 (c) 1 and 12 (d) 1 and 36
3. 3.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 400 (d) 202
4. 4.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)
5. 5.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3035 (c) 3025 (d) 3575
6. 6.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 2 and 10 (c) 5 and 7 (d) 1 and 11
7. 7.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 10404 (c) 9996 (d) 9604
8. 8.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 289 (c) 299 (d) 529
9. 9.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1596 (c) 1454 (d) 1444
10. 10.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 36 (c) 28 (d) 20
11. 11.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 15 (d) 16
12. 12.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
13. 13.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 1 (d) 361
14. 14.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3604 (c) 3844 (d) 3364
15. 15.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 45 (c) 63 (d) 117
16. 16.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4499 (c) 4891 (d) 5329
17. 17.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 6 (c) 256 (d) 36
18. 18.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 196 (c) 151 (d) 106
19. 19.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 7) (c) (x + 1)(x + 12) (d) (x − 2)(x − 6)
20. 20.What is the sum of the roots of x² − 13x + 36 = 0? (a) 13 (b) 14 (c) 12 (d) 36
21. 21.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 1 (c) 169 (d) 225
22. 22.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
23. 23.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 56 (d) 55
24. 24.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 1 (c) 61 (d) 91
25. 25.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 72 (d) 1 and 16
26. 26.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 15 (c) 48 (d) 14
27. 27.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 36 (c) 6 (d) 180
28. 28.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 72 (d) 73
29. 29.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 484 (c) 324 (d) 164
30. 30.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 136 (c) 16 (d) 4
31. 31.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
32. 32.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 3 and 21 (c) 1 and 63 (d) 7 and 9
33. 33.If a + b = 11 and ab = 24, what is a² + b²? (a) 73 (b) 169 (c) 121 (d) 97
34. 34.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 169 (c) 85 (d) 127
35. 35.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 18 (d) 16
36. 36.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 11 (c) 12 (d) 35
37. 37.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x − 3)(x − 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
38. 38.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
39. 39.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)
40. 40.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 7 and 8 (d) 1 and 56

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