Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 869 · 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 + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
2. 2.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2116 (c) 2484 (d) 2916
3. 3.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 225 (c) 441 (d) 657
4. 4.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
5. 5.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3596 (c) 3374 (d) 3364
6. 6.If a + b = 12 and ab = 11, what is a² + b²? (a) 144 (b) 122 (c) 133 (d) 166
7. 7.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4891 (c) 4489 (d) 4499
8. 8.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 16 (d) 14
9. 9.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 40 (c) 64 (d) 52
10. 10.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
11. 11.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 16 (c) 17 (d) 10
12. 12.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 7225 (c) 7235 (d) 8075
13. 13.If a + b = 16 and ab = 55, what is (a − b)²? (a) 6 (b) 256 (c) 36 (d) 146
14. 14.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x − 4)(x − 6) (c) (x + 1)(x + 9) (d) (x + 1)(x + 24)
15. 15.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 3 (c) 9 (d) 361
16. 16.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 1)(x + 9) (c) (x + 2)(x + 8) (d) (x − 2)(x − 8)
17. 17.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 56 (d) 15
18. 18.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x + 1)(x + 27) (d) (x − 3)(x − 9)
19. 19.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 9419 (c) 9409 (d) 10609
20. 20.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 24 (c) 11 (d) 10
21. 21.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 676 (c) 1156 (d) 686
22. 22.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9811 (c) 10201 (d) 9999
23. 23.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 184 (c) 104 (d) 124
24. 24.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)
25. 25.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)
26. 26.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 1 and 10 (c) 4 and 7 (d) 1 and 28
27. 27.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 100 (c) 52 (d) 76
28. 28.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 202 (c) 2 (d) 400
29. 29.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 1 (c) 225 (d) 113
30. 30.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 71 (d) 73
31. 31.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 337 (c) 113 (d) 169
32. 32.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
33. 33.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 193 (c) 169 (d) 157
34. 34.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 7225 (c) 5635 (d) 5625
35. 35.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 36 (c) 35 (d) 34
36. 36.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x − 3)(x − 5)
37. 37.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 5 and 6 (c) 2 and 15 (d) 1 and 10
38. 38.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 196 (c) 16 (d) 4
39. 39.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)
40. 40.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 676 (c) 916 (d) 1156

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