Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 695 · 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 = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 169 (c) 225 (d) 113
2. 2.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 8 (c) 7 (d) 15
3. 3.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 225 (c) 9 (d) 117
4. 4.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 10 (c) 16 (d) 9
5. 5.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 396 (c) 334 (d) 484
6. 6.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 12 (c) 13 (d) 11
7. 7.If a + b = 12 and ab = 27, what is (a − b)²? (a) 6 (b) 36 (c) 90 (d) 144
8. 8.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)
9. 9.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 256 (c) 366 (d) 201
10. 10.What is the product of the roots of x² − 8x + 12 = 0? (a) 13 (b) 11 (c) 12 (d) 8
11. 11.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 40 (c) 12 (d) 13
12. 12.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 1625 (c) 2035 (d) 1225
13. 13.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 3 and 15 (c) 5 and 9 (d) 1 and 45
14. 14.If a + b = 3 and ab = 2, what is a² + b²? (a) 7 (b) 9 (c) 13 (d) 5
15. 15.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 5051 (c) 4761 (d) 4901
16. 16.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 2)(x + 5)
17. 17.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
18. 18.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 1235 (c) 1225 (d) 2025
19. 19.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 42 (c) 13 (d) 14
20. 20.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
21. 21.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 12 (c) 13 (d) 27
22. 22.If a + b = 13 and ab = 40, what is (a − b)²? (a) 9 (b) 89 (c) 3 (d) 169
23. 23.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 85 (c) 127 (d) 253
24. 24.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8464 (c) 8474 (d) 8104
25. 25.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 12 (c) 10 (d) 18
26. 26.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 170 (c) 4 (d) 16
27. 27.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 16 (d) 15
28. 28.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
29. 29.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3035 (c) 3575 (d) 3025
30. 30.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1521 (c) 1599 (d) 1681
31. 31.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 41 (c) 1 (d) 61
32. 32.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1235 (c) 1225 (d) 625
33. 33.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
34. 34.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 1 (d) 289
35. 35.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 266 (c) 384 (d) 576
36. 36.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3969 (b) 3979 (c) 3609 (d) 3249
37. 37.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 4235 (c) 3625 (d) 4225
38. 38.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 304 (c) 592 (d) 400
39. 39.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 2 and 27 (c) 1 and 14 (d) 1 and 54
40. 40.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 121 (c) 93 (d) 177

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