Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 567 · 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 + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)
2. 2.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
3. 3.What is the product of the roots of x² − 11x + 28 = 0? (a) 28 (b) 11 (c) 27 (d) 29
4. 4.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 256 (c) 16 (d) 136
5. 5.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 5041 (c) 5051 (d) 4761
6. 6.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
7. 7.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 5486 (c) 4356 (d) 4916
8. 8.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8116 (c) 8836 (d) 8846
9. 9.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 1 and 16 (d) 8 and 9
10. 10.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 196 (c) 156 (d) 116
11. 11.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 117 (c) 90 (d) 198
12. 12.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)
13. 13.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 15 (d) 17
14. 14.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
16. 16.If a + b = 8 and ab = 12, what is (a − b)²? (a) 16 (b) 4 (c) 64 (d) 40
17. 17.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 45 (c) 81 (d) 9
18. 18.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 81 (c) 1 (d) 61
19. 19.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 55 (d) 56
20. 20.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6401 (c) 6571 (d) 6561
21. 21.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
22. 22.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
23. 23.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 30 (c) 11 (d) 31
24. 24.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 225 (c) 325 (d) 175
25. 25.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 201 (c) 366 (d) 146
26. 26.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 30 (c) 1 and 10 (d) 2 and 15
27. 27.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 117 (c) 3 (d) 225
28. 28.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 125 (c) 25 (d) 5
29. 29.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 65 (c) 93 (d) 177
30. 30.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 52 (c) 88 (d) 40
31. 31.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 3 and 21 (c) 7 and 9 (d) 1 and 63
32. 32.What is the product of the roots of x² − 13x + 40 = 0? (a) 40 (b) 41 (c) 13 (d) 39
33. 33.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 901 (c) 961 (d) 971
34. 34.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 289 (c) 529 (d) 299
35. 35.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 196 (c) 106 (d) 151
36. 36.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 24) (c) (x + 1)(x + 10) (d) (x + 3)(x + 8)
37. 37.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 8125 (c) 7225 (d) 9035
38. 38.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 5 (c) 361 (d) 193
39. 39.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
40. 40.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 144 (c) 122 (d) 133

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