Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 274 · 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 product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 23 (c) 25 (d) 24
2. 2.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 7225 (c) 8075 (d) 9025
3. 3.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 6425 (c) 7225 (d) 5625
4. 4.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 273 (c) 537 (d) 185
5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
6. 6.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 14 (d) 15
7. 7.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 72 (d) 73
8. 8.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 635 (c) 1225 (d) 625
9. 9.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5625 (c) 4925 (d) 5635
10. 10.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
11. 11.What is the sum of the roots of x² − 11x + 28 = 0? (a) 11 (b) 12 (c) 28 (d) 10
12. 12.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
13. 13.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
14. 14.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 11025 (c) 9025 (d) 11035
15. 15.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)
16. 16.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4235 (c) 4225 (d) 4875
17. 17.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1936 (c) 1296 (d) 1616
18. 18.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 1)(x + 28)
19. 19.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 1625 (c) 2035 (d) 1225
20. 20.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 41 (c) 121 (d) 81
21. 21.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2704 (c) 2304 (d) 2496
22. 22.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 784 (c) 1034 (d) 1024
23. 23.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 25 (c) 49 (d) 37
24. 24.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 13 (c) 32 (d) 11
25. 25.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
26. 26.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
27. 27.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 1 (c) 145 (d) 289
28. 28.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1764 (c) 1454 (d) 1444
29. 29.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 891 (c) 1089 (d) 729
30. 30.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 2 and 8 (c) 1 and 21 (d) 1 and 9
31. 31.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10004 (c) 9604 (d) 10414
32. 32.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
33. 33.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 14 (c) 15 (d) 48
34. 34.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 49 (c) 289 (d) 7
35. 35.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 142 (c) 100 (d) 79
36. 36.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 100 (c) 16 (d) 4
37. 37.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 16 (d) 63
38. 38.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
39. 39.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 45) (c) (x + 1)(x + 13) (d) (x − 5)(x − 9)
40. 40.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 784 (c) 794 (d) 896

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