Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 868 · 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 = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 7 (c) 137 (d) 49
2. 2.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 22 (c) 16 (d) 13
3. 3.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)
4. 4.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 35 (c) 11 (d) 12
5. 5.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 53 (c) 54 (d) 55
6. 6.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9984 (c) 9216 (d) 10816
7. 7.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 4 (c) 2 (d) 144
8. 8.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 10009 (c) 10619 (d) 9409
9. 9.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
10. 10.If a + b = 9 and ab = 14, what is (a − b)²? (a) 81 (b) 25 (c) 5 (d) 53
11. 11.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 13 (c) 14 (d) 42
12. 12.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
13. 13.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 2 and 4 (c) 1 and 4 (d) 2 and 3
14. 14.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 12 (c) 30 (d) 10
15. 15.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 149 (c) 429 (d) 219
16. 16.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)
17. 17.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 256 (c) 2 (d) 4
18. 18.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 5486 (c) 4916 (d) 4356
19. 19.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3844 (c) 3364 (d) 3596
20. 20.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 12 (c) 13 (d) 14
21. 21.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 16 (c) 17 (d) 10
22. 22.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
23. 23.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 35 (c) 36 (d) 12
24. 24.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
25. 25.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 2035 (c) 2025 (d) 3025
26. 26.What are the roots of x² − 9x + 14 = 0? (a) 1 and 14 (b) 1 and 8 (c) 3 and 6 (d) 2 and 7
27. 27.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
28. 28.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 8091 (c) 7579 (d) 7569
29. 29.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
30. 30.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 3 and 7 (c) 1 and 9 (d) 2 and 8
31. 31.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 21 (c) 11 (d) 9
32. 32.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 896 (c) 784 (d) 794
33. 33.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 16 (d) 14
34. 34.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3146 (c) 3584 (d) 4096
35. 35.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1936 (d) 1616
36. 36.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 19 (c) 17 (d) 11
37. 37.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 1225 (c) 2035 (d) 1625
38. 38.If a + b = 11 and ab = 18, what is a² + b²? (a) 103 (b) 85 (c) 121 (d) 157
39. 39.If a + b = 12 and ab = 11, what is a² + b²? (a) 122 (b) 133 (c) 166 (d) 144
40. 40.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 169 (c) 129 (d) 89

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