Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 404 · 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 = 14 and ab = 33, what is a² + b²? (a) 262 (b) 163 (c) 196 (d) 130
2. 2.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 6 (c) 100 (d) 36
3. 3.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 14 (c) 40 (d) 13
4. 4.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 202 (c) 400 (d) 2
5. 5.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 64 (c) 6 (d) 36
6. 6.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
7. 7.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)
8. 8.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
9. 9.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 10 (c) 12 (d) 30
10. 10.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6899 (c) 6889 (d) 5929
11. 11.If a + b = 8 and ab = 12, what is (a − b)²? (a) 4 (b) 40 (c) 16 (d) 64
12. 12.If a + b = 17 and ab = 60, what is a² + b²? (a) 169 (b) 409 (c) 229 (d) 289
13. 13.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 121 (c) 61 (d) 81
14. 14.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 9419 (c) 10609 (d) 9409
15. 15.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 10016 (c) 10816 (d) 9216
16. 16.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 15 (d) 56
17. 17.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6084 (c) 6724 (d) 6094
18. 18.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 16 (d) 72
19. 19.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
20. 20.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 157 (c) 5 (d) 25
21. 21.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2501 (c) 2601 (d) 2401
22. 22.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 1225 (c) 1235 (d) 2025
23. 23.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
24. 24.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 529 (c) 289 (d) 391
25. 25.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 149 (c) 289 (d) 219
26. 26.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
27. 27.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
28. 28.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
29. 29.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 1)(x + 24) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
30. 30.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
31. 31.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
32. 32.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 29 (c) 69 (d) 39
33. 33.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 9 (c) 121 (d) 3
34. 34.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 4)(x + 7) (c) (x − 4)(x − 7) (d) (x + 1)(x + 10)
35. 35.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1599 (c) 1531 (d) 1681
36. 36.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 193 (c) 145 (d) 169
37. 37.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 9 (c) 289 (d) 3
38. 38.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 4)(x + 8) (c) (x + 1)(x + 32) (d) (x + 1)(x + 11)
39. 39.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7931 (c) 8099 (d) 7921
40. 40.If a + b = 9 and ab = 8, what is (a − b)²? (a) 7 (b) 49 (c) 65 (d) 81

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