Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 656 · 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 = 60, what is a² + b²? (a) 376 (b) 196 (c) 136 (d) 256
2. 2.If a + b = 15 and ab = 36, what is a² + b²? (a) 189 (b) 225 (c) 153 (d) 297
3. 3.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 971 (c) 841 (d) 961
4. 4.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 14 (c) 15 (d) 16
5. 5.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 32 (c) 33 (d) 31
6. 6.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 10 (c) 11 (d) 28
7. 7.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
8. 8.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 3 and 5 (c) 2 and 6 (d) 1 and 7
9. 9.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 103 (c) 157 (d) 85
10. 10.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
11. 11.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 592 (c) 208 (d) 400
12. 12.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 286 (c) 151 (d) 196
13. 13.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 4901 (c) 5051 (d) 5041
14. 14.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 5635 (c) 4925 (d) 4225
15. 15.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 5635 (c) 5625 (d) 7225
16. 16.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 12 (c) 11 (d) 35
17. 17.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)
18. 18.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
19. 19.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 149 (c) 289 (d) 219
20. 20.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x − 5)(x − 6) (d) (x + 1)(x + 30)
21. 21.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 121 (c) 9 (d) 65
22. 22.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7569 (c) 7579 (d) 8091
23. 23.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 416 (d) 576
24. 24.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 6 (c) 324 (d) 36
25. 25.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2035 (c) 2025 (d) 1625
26. 26.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 63 (c) 1 and 15 (d) 7 and 9
27. 27.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 144 (c) 16 (d) 4
28. 28.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1681 (c) 1599 (d) 1531
29. 29.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 361 (c) 277 (d) 193
30. 30.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 14 (c) 10 (d) 9
31. 31.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
32. 32.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3364 (c) 3596 (d) 3374
33. 33.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 5476 (c) 4356 (d) 4916
34. 34.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
35. 35.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)
36. 36.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 1 and 21 (c) 3 and 7 (d) 1 and 9
37. 37.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)
38. 38.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 12 (c) 8 (d) 7
39. 39.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
40. 40.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 11 (c) 28 (d) 27

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