Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 832 · 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 = 11 and ab = 30, what is a² + b²? (a) 121 (b) 91 (c) 61 (d) 181
2. 2.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 1)(x + 24) (d) (x + 3)(x + 8)
3. 3.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
4. 4.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 16 (c) 17 (d) 15
5. 5.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 196 (c) 156 (d) 276
6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
7. 7.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x + 1)(x + 18) (d) (x − 3)(x − 6)
8. 8.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 324 (c) 484 (d) 244
9. 9.What are the roots of x² − 7x + 12 = 0? (a) 1 and 12 (b) 1 and 6 (c) 3 and 4 (d) 2 and 6
10. 10.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 65 (c) 121 (d) 177
11. 11.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 2035 (c) 1625 (d) 1225
12. 12.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 151 (c) 286 (d) 196
13. 13.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 9614 (c) 9996 (d) 10404
14. 14.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 529 (c) 539 (d) 409
15. 15.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x + 2)(x + 8) (d) (x − 2)(x − 8)
16. 16.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6094 (c) 6724 (d) 6084
17. 17.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 20 (c) 36 (d) 2
18. 18.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4901 (c) 4761 (d) 5041
19. 19.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 121 (c) 9 (d) 81
20. 20.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 35 (c) 13 (d) 11
21. 21.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 1 and 10 (c) 3 and 6 (d) 2 and 9
22. 22.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
23. 23.What are the roots of x² − 9x + 14 = 0? (a) 2 and 7 (b) 1 and 8 (c) 1 and 14 (d) 3 and 6
24. 24.If a + b = 14 and ab = 48, what is a² + b²? (a) 100 (b) 292 (c) 196 (d) 148
25. 25.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 121 (c) 61 (d) 91
26. 26.If a + b = 9 and ab = 8, what is (a − b)²? (a) 49 (b) 65 (c) 81 (d) 7
27. 27.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
28. 28.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 18 (c) 8 (d) 9
29. 29.If a + b = 16 and ab = 55, what is (a − b)²? (a) 6 (b) 36 (c) 146 (d) 256
30. 30.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3481 (c) 3731 (d) 3721
31. 31.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 25 (c) 225 (d) 125
32. 32.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 1 and 16 (d) 8 and 9
33. 33.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 56 (c) 2 and 28 (d) 1 and 14
34. 34.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x − 2)(x − 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
35. 35.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 1)(x + 32)
36. 36.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3491 (c) 3599 (d) 3481
37. 37.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6724 (c) 6084 (d) 6404
38. 38.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
39. 39.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 36 (c) 68 (d) 100
40. 40.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1156 (d) 1166

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