Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 660 · 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 − 14x + 48 = 0? (a) 6 and 8 (b) 1 and 13 (c) 2 and 24 (d) 1 and 48
2. 2.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
3. 3.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 2)(x + 8) (c) (x + 1)(x + 9) (d) (x + 1)(x + 16)
4. 4.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 256 (c) 201 (d) 366
5. 5.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 225 (d) 333
6. 6.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 169 (d) 1
7. 7.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 1)(x + 18) (d) (x + 2)(x + 9)
8. 8.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 8104 (c) 8474 (d) 7744
9. 9.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
10. 10.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
11. 11.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 16 (c) 15 (d) 63
12. 12.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 1 and 14 (c) 2 and 27 (d) 6 and 9
13. 13.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
14. 14.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 1 (c) 121 (d) 91
15. 15.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 7 and 8 (c) 1 and 56 (d) 2 and 28
16. 16.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 9409 (c) 9991 (d) 10609
17. 17.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x + 1)(x + 40) (d) (x − 5)(x − 8)
18. 18.What is the product of the roots of x² − 10x + 21 = 0? (a) 21 (b) 20 (c) 22 (d) 10
19. 19.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 181 (d) 271
20. 20.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 6391 (c) 5939 (d) 6889
21. 21.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)
22. 22.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
23. 23.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 3 and 8 (c) 1 and 10 (d) 2 and 12
24. 24.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 1)(x + 24) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
25. 25.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 196 (c) 286 (d) 151
26. 26.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
27. 27.What are the roots of x² − 9x + 18 = 0? (a) 1 and 8 (b) 2 and 9 (c) 3 and 6 (d) 1 and 18
28. 28.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 41 (c) 43 (d) 13
29. 29.If a + b = 7 and ab = 12, what is (a − b)²? (a) 1 (b) 25 (c) 37 (d) 49
30. 30.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 5339 (c) 4489 (d) 5329
31. 31.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
32. 32.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7931 (b) 8281 (c) 7921 (d) 8099
33. 33.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 169 (c) 73 (d) 97
34. 34.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 1099 (c) 909 (d) 729
35. 35.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 256 (c) 196 (d) 376
36. 36.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 5776 (c) 7056 (d) 6384
37. 37.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 16 (c) 15 (d) 14
38. 38.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)
39. 39.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 45 (c) 117 (d) 81
40. 40.What are the roots of x² − 10x + 16 = 0? (a) 2 and 8 (b) 1 and 16 (c) 1 and 9 (d) 3 and 7

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