Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 694 · 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 + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x + 1)(x + 12)
2. 2.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 404 (c) 324 (d) 494
3. 3.What is the product of the roots of x² − 12x + 32 = 0? (a) 32 (b) 12 (c) 31 (d) 33
4. 4.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3721 (c) 3599 (d) 3491
5. 5.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
6. 6.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 157 (c) 5 (d) 25
7. 7.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 17 (c) 9 (d) 18
8. 8.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
9. 9.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
10. 10.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x + 1)(x + 7) (d) (x − 3)(x − 5)
11. 11.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 2025 (c) 1625 (d) 1225
12. 12.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
13. 13.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 529 (c) 289 (d) 539
14. 14.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 94 (c) 34 (d) 64
15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
16. 16.What are the roots of x² − 14x + 45 = 0? (a) 3 and 15 (b) 1 and 45 (c) 5 and 9 (d) 1 and 13
17. 17.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2916 (c) 2516 (d) 2926
18. 18.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 25 (c) 81 (d) 53
19. 19.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 10404 (c) 9604 (d) 9614
20. 20.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 7 and 8 (d) 1 and 56
21. 21.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 10) (c) (x + 4)(x + 7) (d) (x + 1)(x + 28)
22. 22.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3249 (c) 3591 (d) 3259
23. 23.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 8 (c) 6 (d) 12
24. 24.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1521 (c) 1601 (d) 1681
25. 25.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4916 (c) 4356 (d) 5476
26. 26.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
27. 27.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 61 (c) 41 (d) 81
28. 28.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 8091 (c) 8649 (d) 7579
29. 29.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 5 (c) 193 (d) 361
30. 30.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 225 (c) 441 (d) 3
31. 31.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 396 (c) 324 (d) 484
32. 32.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
33. 33.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 65 (c) 3 (d) 121
34. 34.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 21 (c) 19 (d) 20
35. 35.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
36. 36.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 241 (c) 169 (d) 97
37. 37.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
38. 38.If a + b = 14 and ab = 40, what is a² + b²? (a) 156 (b) 196 (c) 116 (d) 276
39. 39.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x − 2)(x − 3)
40. 40.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 4 (c) 34 (d) 64

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