Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 804 · 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 + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
2. 2.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4356 (c) 5476 (d) 4916
3. 3.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7754 (c) 7744 (d) 8096
4. 4.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 17 (d) 16
5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
6. 6.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 925 (c) 625 (d) 1225
7. 7.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 37 (c) 73 (d) 49
8. 8.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 12 (c) 10 (d) 11
9. 9.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 14 (c) 12 (d) 13
10. 10.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 244 (c) 484 (d) 164
11. 11.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2809 (c) 2491 (d) 2209
12. 12.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)
13. 13.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6401 (c) 6571 (d) 6561
14. 14.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 5329 (c) 4499 (d) 4891
15. 15.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
16. 16.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 441 (c) 399 (d) 361
17. 17.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4875 (c) 4235 (d) 5625
18. 18.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
19. 19.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 12 (c) 18 (d) 10
20. 20.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
21. 21.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 68 (c) 132 (d) 100
22. 22.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 891 (c) 729 (d) 1089
23. 23.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2601 (c) 2501 (d) 2611
24. 24.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 3025 (c) 2475 (d) 2035
25. 25.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 208 (c) 400 (d) 592
26. 26.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 8 (c) 7 (d) 6
27. 27.What is the product of the roots of x² − 8x + 15 = 0? (a) 14 (b) 8 (c) 16 (d) 15
28. 28.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 5041 (c) 5051 (d) 4761
29. 29.If a + b = 16 and ab = 55, what is (a − b)²? (a) 6 (b) 36 (c) 256 (d) 146
30. 30.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4106 (c) 4096 (d) 3136
31. 31.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 2 and 6 (c) 1 and 6 (d) 1 and 12
32. 32.If a + b = 13 and ab = 22, what is (a − b)²? (a) 81 (b) 169 (c) 125 (d) 9
33. 33.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 1024 (d) 896
34. 34.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 56 (d) 15
35. 35.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9025 (c) 9035 (d) 9975
36. 36.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 121 (c) 9 (d) 3
37. 37.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 4 (c) 2 (d) 130
38. 38.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 118 (c) 100 (d) 91
39. 39.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x + 2)(x + 9) (d) (x − 2)(x − 9)
40. 40.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)

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