Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 681 · 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 = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 225 (c) 3 (d) 9
2. 2.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 196 (c) 100 (d) 148
3. 3.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 48 (c) 6 and 8 (d) 1 and 13
4. 4.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x − 2)(x − 8) (c) (x + 1)(x + 9) (d) (x + 1)(x + 16)
5. 5.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3969 (c) 3249 (d) 3979
6. 6.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
7. 7.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
8. 8.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6401 (c) 6241 (d) 6571
9. 9.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)
10. 10.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6734 (c) 6404 (d) 6084
11. 11.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 65 (c) 9 (d) 121
12. 12.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)
13. 13.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 56 (d) 55
14. 14.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 20 (c) 9 (d) 19
15. 15.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 13 (c) 11 (d) 27
16. 16.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
17. 17.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 6)
18. 18.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 1099 (c) 729 (d) 909
19. 19.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 289 (c) 529 (d) 539
20. 20.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
21. 21.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1681 (c) 1521 (d) 1691
22. 22.If a + b = 7 and ab = 10, what is a² + b²? (a) 39 (b) 69 (c) 29 (d) 49
23. 23.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3481 (c) 3601 (d) 3731
24. 24.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
25. 25.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 271 (c) 181 (d) 1
26. 26.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 9 (c) 24 (d) 10
27. 27.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 225 (d) 333
28. 28.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 25 (c) 23 (d) 24
29. 29.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5486 (d) 5476
30. 30.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 2035 (c) 3025 (d) 2025
31. 31.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
32. 32.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)
33. 33.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 2 and 4 (c) 1 and 4 (d) 2 and 3
34. 34.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 324 (c) 468 (d) 252
35. 35.If a + b = 8 and ab = 12, what is a² + b²? (a) 52 (b) 64 (c) 40 (d) 88
36. 36.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
37. 37.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 286 (c) 106 (d) 196
38. 38.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 208 (c) 304 (d) 400
39. 39.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 6416 (c) 7066 (d) 7056
40. 40.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 53 (c) 109 (d) 67

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