Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 696 · 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.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
2. 2.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 2 (c) 10 (d) 16
3. 3.If a + b = 15 and ab = 36, what is a² + b²? (a) 153 (b) 297 (c) 225 (d) 189
4. 4.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
5. 5.If a + b = 15 and ab = 44, what is (a − b)²? (a) 49 (b) 225 (c) 137 (d) 7
6. 6.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)
7. 7.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5635 (c) 5625 (d) 4925
8. 8.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 5476 (c) 4366 (d) 4884
9. 9.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 1)(x + 35)
10. 10.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
11. 11.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4761 (c) 4901 (d) 5041
12. 12.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 117 (c) 198 (d) 90
13. 13.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
14. 14.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6734 (c) 6404 (d) 6084
15. 15.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2314 (c) 2704 (d) 2304
16. 16.If a + b = 10 and ab = 9, what is (a − b)²? (a) 8 (b) 100 (c) 64 (d) 82
17. 17.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 76 (c) 52 (d) 148
18. 18.What is the product of the roots of x² − 11x + 28 = 0? (a) 11 (b) 28 (c) 27 (d) 29
19. 19.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 100 (c) 16 (d) 58
20. 20.If a + b = 12 and ab = 11, what is (a − b)²? (a) 144 (b) 122 (c) 10 (d) 100
21. 21.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 18 (c) 10 (d) 9
22. 22.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
23. 23.If a + b = 11 and ab = 10, what is (a − b)²? (a) 121 (b) 9 (c) 81 (d) 101
24. 24.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 113 (c) 225 (d) 1
25. 25.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 10816 (c) 10016 (d) 9216
26. 26.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 5 and 9 (c) 3 and 15 (d) 1 and 13
27. 27.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 144 (c) 6 (d) 36
28. 28.What are the roots of x² − 11x + 24 = 0? (a) 3 and 8 (b) 1 and 24 (c) 1 and 10 (d) 2 and 12
29. 29.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 2 and 8 (c) 1 and 16 (d) 3 and 7
30. 30.If a + b = 8 and ab = 7, what is (a − b)²? (a) 36 (b) 64 (c) 50 (d) 6
31. 31.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 9035 (c) 8125 (d) 9025
32. 32.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 729 (c) 909 (d) 1089
33. 33.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 1)(x + 18)
34. 34.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 100 (c) 91 (d) 82
35. 35.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 4 (c) 2 (d) 144
36. 36.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 17 (c) 18 (d) 11
37. 37.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 164 (d) 324
38. 38.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
39. 39.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
40. 40.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 112 (c) 144 (d) 80

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