Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 553 · 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 − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
2. 2.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)
3. 3.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 7 and 9 (c) 1 and 15 (d) 3 and 21
4. 4.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 136 (c) 196 (d) 376
5. 5.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)
6. 6.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 9226 (c) 10816 (d) 9216
7. 7.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3596 (c) 3364 (d) 3844
8. 8.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 15 (d) 13
9. 9.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 899 (c) 841 (d) 961
10. 10.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 49 (c) 37 (d) 25
11. 11.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1166 (d) 1156
12. 12.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 7 (c) 13 (d) 5
13. 13.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
14. 14.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 4925 (c) 5625 (d) 4225
15. 15.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 3 (c) 9 (d) 185
16. 16.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 8075 (c) 7235 (d) 7225
17. 17.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 81 (c) 45 (d) 63
18. 18.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 14 (c) 13 (d) 15
19. 19.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
20. 20.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 71 (d) 72
21. 21.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 225 (d) 9
22. 22.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 6 and 9 (c) 1 and 54 (d) 1 and 14
23. 23.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 121 (c) 81 (d) 61
24. 24.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6251 (c) 6561 (d) 6241
25. 25.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 2 and 36 (d) 1 and 72
26. 26.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 145 (c) 289 (d) 1
27. 27.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 576 (d) 416
28. 28.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 100 (c) 8 (d) 64
29. 29.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
30. 30.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 400 (c) 16 (d) 208
31. 31.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
32. 32.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 14 (c) 16 (d) 15
33. 33.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 175 (c) 125 (d) 225
34. 34.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 62 (d) 63
35. 35.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 266 (c) 384 (d) 256
36. 36.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 10 (c) 11 (d) 28
37. 37.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)
38. 38.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 961 (d) 841
39. 39.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 896 (c) 1024 (d) 784
40. 40.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 425 (c) 225 (d) 635

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