Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 515 · 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.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 256 (c) 366 (d) 201
2. 2.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 169 (c) 409 (d) 229
3. 3.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
4. 4.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
5. 5.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 5 (c) 25 (d) 289
6. 6.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 1 and 72 (d) 2 and 36
7. 7.What are the roots of x² − 11x + 24 = 0? (a) 3 and 8 (b) 2 and 12 (c) 1 and 24 (d) 1 and 10
8. 8.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 7 and 8 (d) 1 and 56
9. 9.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 52 (c) 4 (d) 2
10. 10.If a + b = 11 and ab = 10, what is (a − b)²? (a) 9 (b) 101 (c) 81 (d) 121
11. 11.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3601 (c) 3721 (d) 3481
12. 12.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 101 (c) 121 (d) 141
13. 13.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 7579 (c) 8091 (d) 8649
14. 14.If a + b = 14 and ab = 24, what is (a − b)²? (a) 100 (b) 196 (c) 10 (d) 148
15. 15.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 2 (c) 10 (d) 16
16. 16.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 3616 (c) 4096 (d) 4106
17. 17.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 8 (c) 7 (d) 9
18. 18.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5939 (c) 6889 (d) 5929
19. 19.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1235 (c) 1575 (d) 1225
20. 20.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)
21. 21.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5194 (c) 5184 (d) 4624
22. 22.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 28 (c) 11 (d) 27
23. 23.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 3 (c) 9 (d) 81
24. 24.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 17 (d) 73
25. 25.What are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 1 and 12 (c) 1 and 36 (d) 4 and 9
26. 26.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 14) (c) (x − 2)(x − 7) (d) (x + 1)(x + 8)
27. 27.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
28. 28.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 2 and 5 (c) 1 and 5 (d) 1 and 8
29. 29.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
30. 30.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 1)(x + 10)
31. 31.If a + b = 13 and ab = 12, what is a² + b²? (a) 193 (b) 169 (c) 157 (d) 145
32. 32.What is the product of the roots of x² − 8x + 15 = 0? (a) 14 (b) 16 (c) 8 (d) 15
33. 33.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 13 (c) 40 (d) 14
34. 34.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 9216 (c) 10016 (d) 10826
35. 35.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 4 (d) 400
36. 36.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 225 (c) 625 (d) 425
37. 37.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 4 (d) 16
38. 38.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 41 (d) 43
39. 39.What is the product of the roots of x² − 8x + 12 = 0? (a) 8 (b) 13 (c) 12 (d) 11
40. 40.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)

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