Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 630 · 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.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
2. 2.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 73 (c) 169 (d) 97
3. 3.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 289 (c) 3 (d) 9
4. 4.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 2 (c) 144 (d) 74
5. 5.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 111 (c) 101 (d) 121
6. 6.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 21 (c) 17 (d) 33
7. 7.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 223 (c) 289 (d) 157
8. 8.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x − 2)(x − 3)
9. 9.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2484 (c) 2116 (d) 2916
10. 10.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 13 (c) 14 (d) 45
11. 11.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
12. 12.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 299 (c) 529 (d) 391
13. 13.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 441 (d) 3
14. 14.If a + b = 9 and ab = 8, what is a² + b²? (a) 73 (b) 81 (c) 65 (d) 97
15. 15.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 404 (c) 324 (d) 494
16. 16.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 286 (c) 151 (d) 196
17. 17.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 5041 (c) 4761 (d) 4899
18. 18.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
19. 19.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 72 (d) 16
20. 20.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3259 (c) 3249 (d) 3591
21. 21.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 145 (d) 217
22. 22.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
23. 23.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2714 (c) 2504 (d) 2704
24. 24.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x − 5)(x − 7)
25. 25.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x + 1)(x + 13) (d) (x − 5)(x − 9)
26. 26.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 16 (c) 208 (d) 4
27. 27.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 11 (c) 12 (d) 13
28. 28.What is the product of the roots of x² − 7x + 10 = 0? (a) 10 (b) 9 (c) 11 (d) 7
29. 29.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 598 (c) 202 (d) 400
30. 30.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1521 (c) 1599 (d) 1531
31. 31.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 9 (c) 5 (d) 7
32. 32.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 73 (d) 17
33. 33.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 8091 (c) 7579 (d) 7569
34. 34.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 16 (c) 256 (d) 4
35. 35.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x − 4)(x − 7)
36. 36.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3249 (c) 3609 (d) 3969
37. 37.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
38. 38.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
39. 39.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4624 (c) 4896 (d) 5184
40. 40.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 49 (c) 289 (d) 169

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