Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 111 · 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 + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
2. 2.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 7921 (c) 8281 (d) 8291
3. 3.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 8075 (c) 7235 (d) 9025
4. 4.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 145 (c) 217 (d) 289
5. 5.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
6. 6.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6084 (c) 6094 (d) 6396
7. 7.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 185 (c) 537 (d) 273
8. 8.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3599 (c) 3721 (d) 3491
9. 9.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 4225 (c) 4235 (d) 3625
10. 10.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 6425 (c) 7225 (d) 7235
11. 11.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 5 and 8 (c) 2 and 20 (d) 1 and 12
12. 12.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 271 (d) 181
13. 13.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 1 and 4 (c) 2 and 3 (d) 1 and 6
14. 14.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
15. 15.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 93 (c) 177 (d) 121
16. 16.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 8 and 9 (d) 1 and 72
17. 17.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 71 (d) 17
18. 18.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 47 (c) 49 (d) 14
19. 19.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 10609 (c) 9409 (d) 9419
20. 20.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
21. 21.What is the product of the roots of x² − 8x + 12 = 0? (a) 13 (b) 11 (c) 12 (d) 8
22. 22.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 25 (c) 5 (d) 73
23. 23.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 625 (c) 235 (d) 225
24. 24.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1444 (c) 1764 (d) 1596
25. 25.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 196 (c) 130 (d) 262
26. 26.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 441 (c) 399 (d) 371
27. 27.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3146 (c) 3136 (d) 3584
28. 28.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
29. 29.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
30. 30.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5776 (c) 7056 (d) 5786
31. 31.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 916 (c) 1156 (d) 676
32. 32.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 361 (c) 3 (d) 9
33. 33.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 539 (c) 289 (d) 409
34. 34.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 40 (c) 12 (d) 14
35. 35.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 225 (c) 333 (d) 441
36. 36.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 1 and 8 (c) 2 and 9 (d) 3 and 6
37. 37.What is the product of the roots of x² − 11x + 28 = 0? (a) 28 (b) 29 (c) 27 (d) 11
38. 38.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
39. 39.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16
40. 40.If a + b = 14 and ab = 40, what is a² + b²? (a) 156 (b) 116 (c) 276 (d) 196

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