Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 854 · 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.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 739 (c) 729 (d) 891
2. 2.What is the product of the roots of x² − 11x + 28 = 0? (a) 28 (b) 29 (c) 27 (d) 11
3. 3.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 1)(x + 11)
4. 4.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 10004 (c) 9604 (d) 10404
5. 5.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 6 and 9 (c) 1 and 54 (d) 2 and 27
6. 6.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 11 (c) 13 (d) 12
7. 7.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
8. 8.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 235 (d) 625
9. 9.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 851 (c) 841 (d) 961
10. 10.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1764 (c) 1604 (d) 1774
11. 11.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 164 (c) 484 (d) 324
12. 12.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
13. 13.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
14. 14.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7744 (c) 8096 (d) 7754
15. 15.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3481 (c) 3731 (d) 3721
16. 16.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5625 (b) 6375 (c) 5635 (d) 7225
17. 17.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
18. 18.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 52 (c) 40 (d) 88
19. 19.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 72 (d) 1 and 16
20. 20.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 3025 (c) 4225 (d) 4235
21. 21.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 217 (c) 145 (d) 289
22. 22.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 11 (c) 12 (d) 18
23. 23.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 11035 (c) 9025 (d) 11025
24. 24.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 784 (c) 904 (d) 1024
25. 25.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6724 (c) 6084 (d) 6396
26. 26.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 4) (c) (x + 1)(x + 6) (d) (x − 2)(x − 3)
27. 27.If a + b = 15 and ab = 44, what is a² + b²? (a) 137 (b) 313 (c) 181 (d) 225
28. 28.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 1)(x + 24)
29. 29.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 48 (c) 47 (d) 49
30. 30.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 16 (c) 2 (d) 10
31. 31.If a + b = 7 and ab = 10, what is (a − b)²? (a) 3 (b) 49 (c) 9 (d) 29
32. 32.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 256 (c) 160 (d) 208
33. 33.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 4901 (c) 4761 (d) 5051
34. 34.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 15 (c) 1 and 63 (d) 7 and 9
35. 35.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
36. 36.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 15 (c) 9 (d) 8
37. 37.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2499 (c) 2401 (d) 2411
38. 38.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6241 (c) 6399 (d) 6251
39. 39.If a + b = 16 and ab = 48, what is (a − b)²? (a) 8 (b) 160 (c) 256 (d) 64
40. 40.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 468 (c) 252 (d) 180

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