Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 384 · 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 = 20 and ab = 99, what is a² + b²? (a) 202 (b) 301 (c) 598 (d) 400
2. 2.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)
3. 3.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 2 and 27 (c) 1 and 14 (d) 1 and 54
4. 4.If a + b = 9 and ab = 14, what is a² + b²? (a) 67 (b) 109 (c) 53 (d) 81
5. 5.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
6. 6.What is the product of the roots of x² − 14x + 45 = 0? (a) 45 (b) 44 (c) 14 (d) 46
7. 7.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 3 (c) 169 (d) 9
8. 8.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 16 (c) 15 (d) 56
9. 9.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 8 and 9 (d) 2 and 36
10. 10.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 2 and 20 (c) 5 and 8 (d) 1 and 12
11. 11.If a + b = 14 and ab = 33, what is a² + b²? (a) 130 (b) 262 (c) 163 (d) 196
12. 12.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3136 (c) 3146 (d) 3584
13. 13.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 125 (c) 325 (d) 175
14. 14.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 4 (c) 2 (d) 400
15. 15.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 15 (c) 3 and 5 (d) 1 and 7
16. 16.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
17. 17.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 2 and 18 (c) 4 and 9 (d) 1 and 12
18. 18.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 1 (c) 145 (d) 217
19. 19.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 324 (c) 170 (d) 247
20. 20.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
21. 21.If a + b = 15 and ab = 54, what is a² + b²? (a) 117 (b) 225 (c) 171 (d) 333
22. 22.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 1089 (c) 739 (d) 891
23. 23.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 100 (c) 68 (d) 132
24. 24.If a + b = 13 and ab = 36, what is a² + b²? (a) 241 (b) 133 (c) 97 (d) 169
25. 25.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
26. 26.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 273 (c) 361 (d) 537
27. 27.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 961 (c) 851 (d) 841
28. 28.If a + b = 13 and ab = 30, what is (a − b)²? (a) 49 (b) 7 (c) 109 (d) 169
29. 29.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 1 and 11 (c) 3 and 9 (d) 2 and 10
30. 30.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4904 (c) 5194 (d) 4624
31. 31.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 6) (c) (x + 1)(x + 10) (d) (x − 2)(x − 5)
32. 32.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 11 (c) 10 (d) 18
33. 33.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
34. 34.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
35. 35.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
36. 36.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 10 (c) 24 (d) 11
37. 37.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
38. 38.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2501 (c) 2401 (d) 2611
39. 39.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 9 (c) 8 (d) 10
40. 40.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 14 (c) 13 (d) 12

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