Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 987 · 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 = 9 and ab = 18, what is a² + b²? (a) 45 (b) 63 (c) 81 (d) 117
2. 2.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
3. 3.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 2)(x + 9) (c) (x + 1)(x + 18) (d) (x − 2)(x − 9)
4. 4.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
5. 5.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 3 (c) 117 (d) 225
6. 6.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2116 (c) 2916 (d) 2926
7. 7.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 142 (c) 100 (d) 58
8. 8.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 625 (c) 1235 (d) 925
9. 9.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 361 (c) 271 (d) 181
10. 10.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 132 (c) 68 (d) 84
11. 11.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 9999 (c) 9801 (d) 10201
12. 12.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
13. 13.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)
14. 14.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
15. 15.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x − 4)(x − 6)
16. 16.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 16 (d) 1 and 72
17. 17.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
18. 18.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 89 (c) 169 (d) 129
19. 19.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 25 (c) 169 (d) 5
20. 20.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 2035 (c) 3025 (d) 2025
21. 21.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 196 (c) 172 (d) 148
22. 22.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 61 (c) 121 (d) 1
23. 23.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 17 (d) 71
24. 24.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 9 (c) 10 (d) 8
25. 25.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 196 (c) 156 (d) 276
26. 26.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3481 (c) 3599 (d) 3491
27. 27.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 15 (d) 57
28. 28.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 1625 (c) 2035 (d) 2025
29. 29.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 891 (c) 739 (d) 729
30. 30.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
31. 31.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 5329 (c) 4499 (d) 4891
32. 32.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5939 (c) 5929 (d) 6889
33. 33.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 841 (c) 971 (d) 901
34. 34.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 2)(x + 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
35. 35.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
36. 36.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 15 (d) 14
37. 37.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 10 (c) 196 (d) 100
38. 38.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
39. 39.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
40. 40.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 169 (c) 225 (d) 1

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