Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 51 · 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.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 225 (c) 1 (d) 113
2. 2.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 2 and 5 (c) 1 and 5 (d) 2 and 4
3. 3.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 15 (d) 13
4. 4.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
5. 5.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 25 (c) 361 (d) 193
6. 6.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3146 (c) 3584 (d) 3136
7. 7.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 19 (c) 9 (d) 20
8. 8.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 396 (c) 324 (d) 334
9. 9.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 8)
10. 10.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 132 (c) 84 (d) 68
11. 11.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 37 (c) 25 (d) 73
12. 12.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 117 (c) 63 (d) 45
13. 13.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
14. 14.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 2 (d) 4
15. 15.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
16. 16.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 635 (c) 225 (d) 425
17. 17.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)
18. 18.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 256 (c) 136 (d) 196
19. 19.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
20. 20.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 586 (d) 576
21. 21.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 2 and 27 (c) 1 and 14 (d) 6 and 9
22. 22.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 4624 (c) 5184 (d) 4904
23. 23.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
24. 24.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 529 (c) 193 (d) 361
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 + 54) (d) (x + 1)(x + 14)
26. 26.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 12 (c) 14 (d) 13
27. 27.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x + 1)(x + 40) (d) (x − 5)(x − 8)
28. 28.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 164 (d) 324
29. 29.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 6 (d) 8
30. 30.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 9025 (c) 11025 (d) 11035
31. 31.What is the sum of the roots of x² − 14x + 45 = 0? (a) 14 (b) 45 (c) 15 (d) 13
32. 32.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)
33. 33.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 149 (c) 3 (d) 289
34. 34.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 262 (c) 196 (d) 130
35. 35.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 5625 (c) 6375 (d) 7225
36. 36.If a + b = 12 and ab = 32, what is a² + b²? (a) 144 (b) 112 (c) 80 (d) 208
37. 37.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 8 (c) 104 (d) 64
38. 38.What is the sum of the roots of x² − 13x + 36 = 0? (a) 13 (b) 14 (c) 36 (d) 12
39. 39.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 4 (c) 2 (d) 324
40. 40.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10404 (c) 10414 (d) 9604

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