Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 550 · 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 + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
2. 2.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)
3. 3.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 841 (d) 961
4. 4.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7744 (c) 8096 (d) 7754
5. 5.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 16 (d) 18
6. 6.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2411 (c) 2601 (d) 2499
7. 7.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 2 and 27 (c) 1 and 14 (d) 1 and 54
8. 8.What is the product of the roots of x² − 13x + 40 = 0? (a) 39 (b) 40 (c) 13 (d) 41
9. 9.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 396 (c) 334 (d) 324
10. 10.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 121 (c) 81 (d) 9
11. 11.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 909 (c) 729 (d) 1099
12. 12.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5329 (b) 4489 (c) 4909 (d) 5339
13. 13.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 1 and 9 (c) 1 and 21 (d) 3 and 7
14. 14.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 4 (c) 2 (d) 64
15. 15.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 14 (d) 12
16. 16.What are the roots of x² − 7x + 12 = 0? (a) 2 and 6 (b) 1 and 12 (c) 1 and 6 (d) 3 and 4
17. 17.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 23 (c) 24 (d) 25
18. 18.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 4 (c) 324 (d) 16
19. 19.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
20. 20.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 1)(x + 16) (d) (x + 2)(x + 8)
21. 21.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 4 (d) 2
22. 22.If a + b = 9 and ab = 8, what is a² + b²? (a) 65 (b) 81 (c) 97 (d) 73
23. 23.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 5786 (c) 5776 (d) 6384
24. 24.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 185 (c) 273 (d) 537
25. 25.If a + b = 10 and ab = 24, what is a² + b²? (a) 52 (b) 76 (c) 100 (d) 148
26. 26.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
27. 27.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 324 (c) 404 (d) 494
28. 28.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 14 (c) 12 (d) 42
29. 29.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 52 (c) 100 (d) 4
30. 30.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2501 (c) 2601 (d) 2401
31. 31.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 2 and 36 (d) 1 and 16
32. 32.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 371 (c) 399 (d) 441
33. 33.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
34. 34.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 625 (c) 235 (d) 375
35. 35.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 3 (c) 149 (d) 9
36. 36.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 9604 (c) 10414 (d) 10404
37. 37.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 15 (c) 54 (d) 53
38. 38.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
39. 39.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 193 (c) 145 (d) 169
40. 40.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 11 (c) 12 (d) 13

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