Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 880 · 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 = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 1 (c) 61 (d) 81
2. 2.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 2 and 36 (d) 1 and 16
3. 3.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 64 (c) 8 (d) 100
4. 4.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)
5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
6. 6.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 7225 (c) 8125 (d) 9025
7. 7.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3604 (c) 3364 (d) 3854
8. 8.What is the product of the roots of x² − 12x + 27 = 0? (a) 12 (b) 26 (c) 28 (d) 27
9. 9.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 12 (c) 13 (d) 7
10. 10.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 37 (c) 49 (d) 25
11. 11.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
12. 12.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 52 (c) 100 (d) 76
13. 13.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 121 (c) 85 (d) 103
14. 14.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 9 (c) 18 (d) 8
15. 15.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 4916 (c) 5476 (d) 5486
16. 16.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 5184 (c) 4634 (d) 4624
17. 17.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 11 (c) 28 (d) 27
18. 18.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 3625 (c) 4225 (d) 3025
19. 19.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 48 (c) 13 (d) 14
20. 20.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1531 (c) 1599 (d) 1681
21. 21.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 676 (c) 1156 (d) 916
22. 22.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
23. 23.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 289 (c) 429 (d) 149
24. 24.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 171 (c) 117 (d) 225
25. 25.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 451 (c) 401 (d) 361
26. 26.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6251 (c) 6241 (d) 6399
27. 27.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
28. 28.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 196 (c) 36 (d) 6
29. 29.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
30. 30.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 109 (c) 144 (d) 214
31. 31.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 4 (d) 400
32. 32.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 12)
33. 33.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 4 (c) 324 (d) 2
34. 34.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2509 (c) 2209 (d) 2819
35. 35.What is the product of the roots of x² − 10x + 21 = 0? (a) 21 (b) 10 (c) 22 (d) 20
36. 36.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 2 and 20 (c) 5 and 8 (d) 1 and 40
37. 37.What is the product of the roots of x² − 8x + 15 = 0? (a) 15 (b) 16 (c) 14 (d) 8
38. 38.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x − 2)(x − 7)
39. 39.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
40. 40.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 1024 (c) 794 (d) 784

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