Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 323 · 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.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 11 (c) 35 (d) 12
2. 2.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 676 (c) 1166 (d) 916
3. 3.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 256 (c) 193 (d) 382
4. 4.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 4225 (c) 5635 (d) 4925
5. 5.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 61 (c) 37 (d) 43
6. 6.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 961 (c) 971 (d) 841
7. 7.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 11)
8. 8.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 8 (c) 9 (d) 10
9. 9.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4891 (c) 4489 (d) 4499
10. 10.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6401 (c) 6571 (d) 6561
11. 11.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 225 (c) 9 (d) 441
12. 12.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 10 (c) 11 (d) 18
13. 13.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 539 (c) 409 (d) 529
14. 14.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 10 (c) 9 (d) 18
15. 15.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 7 (c) 6 (d) 8
16. 16.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 11 (c) 30 (d) 10
17. 17.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4901 (c) 4761 (d) 5041
18. 18.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)
19. 19.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 17 (c) 63 (d) 16
20. 20.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 1 (d) 289
21. 21.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 2 and 14 (c) 4 and 7 (d) 1 and 10
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 sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 15 (c) 13 (d) 48
24. 24.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 10) (c) (x + 1)(x + 6) (d) (x − 2)(x − 5)
25. 25.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 451 (d) 441
26. 26.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 657 (c) 333 (d) 441
27. 27.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 5625 (c) 7225 (d) 6425
28. 28.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
29. 29.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 18 (c) 17 (d) 11
30. 30.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
31. 31.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 145 (c) 433 (d) 217
32. 32.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 1024 (c) 784 (d) 896
33. 33.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 16 (c) 15 (d) 56
34. 34.If a + b = 12 and ab = 32, what is (a − b)²? (a) 144 (b) 16 (c) 4 (d) 80
35. 35.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)
36. 36.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 40) (c) (x + 1)(x + 12) (d) (x − 5)(x − 8)
37. 37.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
38. 38.If a + b = 11 and ab = 24, what is (a − b)²? (a) 5 (b) 73 (c) 25 (d) 121
39. 39.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 252 (c) 324 (d) 180
40. 40.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 148 (c) 76 (d) 52

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