Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 663 · 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 + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
2. 2.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3601 (c) 3731 (d) 3481
3. 3.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 289 (c) 9 (d) 149
4. 4.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 9 (c) 121 (d) 65
5. 5.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 256 (c) 130 (d) 382
6. 6.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 289 (c) 391 (d) 299
7. 7.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 10 (c) 24 (d) 11
8. 8.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)
9. 9.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4499 (c) 5329 (d) 4891
10. 10.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 100 (c) 2 (d) 4
11. 11.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 625 (c) 375 (d) 235
12. 12.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
13. 13.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 5051 (c) 5041 (d) 4761
14. 14.What are the roots of x² − 9x + 18 = 0? (a) 1 and 8 (b) 2 and 9 (c) 1 and 18 (d) 3 and 6
15. 15.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 40)
16. 16.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 361 (c) 399 (d) 371
17. 17.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1936 (d) 1616
18. 18.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 441 (c) 401 (d) 451
19. 19.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
20. 20.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)
21. 21.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
22. 22.If a + b = 10 and ab = 24, what is a² + b²? (a) 52 (b) 148 (c) 100 (d) 76
23. 23.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 494 (d) 484
24. 24.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 676 (c) 916 (d) 1156
25. 25.If a + b = 8 and ab = 12, what is (a − b)²? (a) 4 (b) 16 (c) 40 (d) 64
26. 26.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
27. 27.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 25 (c) 5 (d) 225
28. 28.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 904 (c) 1034 (d) 1024
29. 29.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 541 (d) 271
30. 30.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4106 (c) 4096 (d) 3136
31. 31.If a + b = 14 and ab = 33, what is (a − b)²? (a) 196 (b) 130 (c) 64 (d) 8
32. 32.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 153 (c) 9 (d) 225
33. 33.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
34. 34.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 16 (c) 63 (d) 15
35. 35.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10201 (c) 10211 (d) 9801
36. 36.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 74 (c) 2 (d) 4
37. 37.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
38. 38.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 2 (c) 20 (d) 36
39. 39.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 202 (c) 301 (d) 598
40. 40.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 1235 (c) 2025 (d) 1225

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