Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 608 · 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.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 4 and 7 (c) 1 and 28 (d) 1 and 10
2. 2.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 8 and 9 (d) 1 and 72
3. 3.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 42 (d) 1 and 12
4. 4.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 14 (d) 1 and 56
5. 5.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
6. 6.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 43 (d) 41
7. 7.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 79 (c) 58 (d) 100
8. 8.If a + b = 8 and ab = 12, what is a² + b²? (a) 52 (b) 40 (c) 64 (d) 88
9. 9.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 484 (c) 494 (d) 404
10. 10.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 1 (c) 85 (d) 169
11. 11.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 15 (d) 16
12. 12.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 6 and 8 (c) 1 and 48 (d) 2 and 24
13. 13.What is the product of the roots of x² − 8x + 12 = 0? (a) 11 (b) 8 (c) 12 (d) 13
14. 14.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 5 (c) 9 (d) 1
15. 15.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
16. 16.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
17. 17.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 9025 (c) 11035 (d) 10025
18. 18.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 337 (c) 225 (d) 113
19. 19.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)
20. 20.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1625 (c) 1225 (d) 2025
21. 21.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 576 (c) 384 (d) 266
22. 22.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 49 (c) 25 (d) 1
23. 23.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 1 and 12 (c) 2 and 6 (d) 1 and 6
24. 24.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8109 (c) 7569 (d) 8659
25. 25.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 361 (c) 273 (d) 185
26. 26.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 441 (c) 399 (d) 361
27. 27.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 63 (d) 16
28. 28.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 35) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
29. 29.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 74 (c) 109 (d) 144
30. 30.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
31. 31.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2601 (c) 2611 (d) 2401
32. 32.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2401 (c) 2601 (d) 2411
33. 33.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 202 (c) 598 (d) 301
34. 34.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 68 (c) 100 (d) 84
35. 35.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x − 2)(x − 7)
36. 36.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4489 (c) 4909 (d) 5329
37. 37.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)
38. 38.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 25 (c) 24 (d) 10
39. 39.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 13 (c) 37 (d) 25
40. 40.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x − 5)(x − 6) (d) (x + 1)(x + 10)

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