Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 154 · 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 = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 289 (c) 217 (d) 145
2. 2.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10404 (c) 10004 (d) 10414
3. 3.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)
4. 4.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16
5. 5.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 164 (c) 484 (d) 324
6. 6.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1599 (c) 1681 (d) 1531
7. 7.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 3 and 7 (c) 1 and 9 (d) 2 and 8
8. 8.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 11 (c) 16 (d) 10
9. 9.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 361 (c) 441 (d) 371
10. 10.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 1 (c) 41 (d) 61
11. 11.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 273 (c) 361 (d) 537
12. 12.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 8084 (c) 8836 (d) 7406
13. 13.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8291 (c) 7921 (d) 8281
14. 14.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 21 (c) 22 (d) 20
15. 15.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
16. 16.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2035 (c) 2025 (d) 1625
17. 17.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 12 (c) 14 (d) 13
18. 18.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 334 (c) 396 (d) 484
19. 19.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 4 and 5 (c) 1 and 20 (d) 1 and 8
20. 20.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 196 (c) 276 (d) 156
21. 21.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 225 (d) 333
22. 22.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
23. 23.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
24. 24.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 400 (c) 304 (d) 592
25. 25.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 20 (c) 9 (d) 10
26. 26.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 5 (c) 6 (d) 7
27. 27.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 28 (c) 11 (d) 27
28. 28.If a + b = 15 and ab = 36, what is a² + b²? (a) 153 (b) 189 (c) 225 (d) 297
29. 29.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
30. 30.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 5929 (c) 6899 (d) 6889
31. 31.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 74 (c) 214 (d) 109
32. 32.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
33. 33.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 400 (d) 4
34. 34.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 63 (d) 16
35. 35.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
36. 36.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 6 and 9 (c) 1 and 14 (d) 2 and 27
37. 37.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 256 (c) 384 (d) 266
38. 38.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 46 (c) 14 (d) 45
39. 39.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 56 (d) 55
40. 40.What are the roots of x² − 11x + 24 = 0? (a) 3 and 8 (b) 1 and 10 (c) 1 and 24 (d) 2 and 12

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