Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 902 · 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 = 18 and ab = 80, what is a² + b²? (a) 484 (b) 164 (c) 324 (d) 244
2. 2.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
3. 3.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 304 (c) 592 (d) 400
4. 4.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 185 (c) 361 (d) 3
5. 5.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 196 (c) 106 (d) 4
6. 6.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 7569 (c) 8649 (d) 8659
7. 7.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 9025 (c) 11025 (d) 11035
8. 8.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 74 (c) 2 (d) 144
9. 9.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
10. 10.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 9025 (c) 7225 (d) 8125
11. 11.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 16 (d) 18
12. 12.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 44 (c) 46 (d) 45
13. 13.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
14. 14.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 253 (c) 85 (d) 127
15. 15.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4096 (c) 3616 (d) 4106
16. 16.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
17. 17.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 24 (c) 23 (d) 10
18. 18.If a + b = 8 and ab = 12, what is a² + b²? (a) 52 (b) 40 (c) 64 (d) 88
19. 19.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
20. 20.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 9801 (d) 10001
21. 21.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 8096 (c) 7754 (d) 8464
22. 22.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 171 (c) 225 (d) 117
23. 23.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
24. 24.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 686 (c) 884 (d) 676
25. 25.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 7 (c) 49 (d) 137
26. 26.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 13 (c) 12 (d) 32
27. 27.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10414 (c) 10004 (d) 10404
28. 28.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 277 (c) 361 (d) 193
29. 29.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 1 and 24 (c) 4 and 6 (d) 2 and 12
30. 30.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 10 (d) 9
31. 31.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x − 3)(x − 4)
32. 32.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1369 (c) 1591 (d) 1379
33. 33.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 225 (c) 9 (d) 81
34. 34.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 9 (c) 3 (d) 225
35. 35.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3481 (c) 3599 (d) 3491
36. 36.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
37. 37.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)
38. 38.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 6425 (c) 7235 (d) 7225
39. 39.What is the product of the roots of x² − 10x + 16 = 0? (a) 17 (b) 16 (c) 15 (d) 10
40. 40.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 262 (c) 163 (d) 130

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