Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 185 · 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 28²? (a) 794 (b) 784 (c) 1024 (d) 896
2. 2.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 1 and 14 (d) 7 and 8
3. 3.If a + b = 9 and ab = 14, what is (a − b)²? (a) 81 (b) 25 (c) 5 (d) 53
4. 4.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 1 (c) 225 (d) 113
5. 5.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
6. 6.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 12 (c) 14 (d) 13
7. 7.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10201 (c) 10211 (d) 9801
8. 8.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 366 (c) 256 (d) 146
9. 9.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 1)(x + 14) (d) (x + 2)(x + 7)
10. 10.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)
11. 11.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)
12. 12.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 45 (c) 14 (d) 46
13. 13.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 54 (c) 2 and 27 (d) 1 and 14
14. 14.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 17 (c) 16 (d) 15
15. 15.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 4 (c) 40 (d) 16
16. 16.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 4771 (c) 5041 (d) 4761
17. 17.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)
18. 18.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 225 (d) 333
19. 19.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 884 (c) 676 (d) 686
20. 20.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)
21. 21.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 116 (c) 276 (d) 156
22. 22.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 9 (c) 7 (d) 8
23. 23.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 91 (c) 121 (d) 1
24. 24.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 401 (c) 441 (d) 361
25. 25.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)
26. 26.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 2025 (c) 1575 (d) 1235
27. 27.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
28. 28.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 6384 (c) 7056 (d) 5776
29. 29.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 7931 (c) 8281 (d) 8099
30. 30.If a + b = 16 and ab = 48, what is a² + b²? (a) 208 (b) 256 (c) 352 (d) 160
31. 31.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 164 (c) 4 (d) 2
32. 32.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 36 (c) 34 (d) 35
33. 33.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 4235 (c) 4225 (d) 5625
34. 34.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 3 (c) 45 (d) 9
35. 35.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3604 (c) 3364 (d) 3844
36. 36.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 2035 (c) 3025 (d) 2025
37. 37.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6561 (c) 6241 (d) 6401
38. 38.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 169 (c) 85 (d) 127
39. 39.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)
40. 40.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 4225 (c) 5635 (d) 5625

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