Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 603 · 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.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 7225 (c) 9025 (d) 8075
2. 2.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 2)(x + 8) (c) (x + 1)(x + 9) (d) (x − 2)(x − 8)
3. 3.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 4 (d) 16
4. 4.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 225 (c) 657 (d) 441
5. 5.If a + b = 12 and ab = 27, what is (a − b)²? (a) 36 (b) 144 (c) 6 (d) 90
6. 6.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2401 (c) 2499 (d) 2411
7. 7.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3844 (c) 3604 (d) 3364
8. 8.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 12 (c) 13 (d) 27
9. 9.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2611 (c) 2501 (d) 2401
10. 10.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3481 (c) 3601 (d) 3731
11. 11.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 193 (c) 529 (d) 277
12. 12.If a + b = 14 and ab = 40, what is a² + b²? (a) 156 (b) 276 (c) 196 (d) 116
13. 13.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 5786 (c) 6384 (d) 5776
14. 14.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 425 (c) 635 (d) 625
15. 15.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 109 (c) 74 (d) 144
16. 16.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1575 (c) 1225 (d) 1235
17. 17.What is the product of the roots of x² − 9x + 18 = 0? (a) 17 (b) 19 (c) 9 (d) 18
18. 18.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3035 (c) 3025 (d) 2025
19. 19.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 14 (c) 48 (d) 15
20. 20.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)
21. 21.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 10 (c) 25 (d) 24
22. 22.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 1)(x + 12) (d) (x + 5)(x + 8)
23. 23.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 64 (c) 8 (d) 100
24. 24.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 5 (c) 7 (d) 9
25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
26. 26.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6724 (c) 6404 (d) 6084
27. 27.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 1)(x + 24) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
28. 28.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 100 (c) 68 (d) 132
29. 29.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 8 (c) 12 (d) 7
30. 30.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
31. 31.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 1 and 9 (c) 2 and 12 (d) 4 and 6
32. 32.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 24 (c) 11 (d) 10
33. 33.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
34. 34.If a + b = 14 and ab = 33, what is (a − b)²? (a) 130 (b) 8 (c) 196 (d) 64
35. 35.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2484 (c) 2126 (d) 2116
36. 36.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 21 (c) 10 (d) 20
37. 37.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 25 (c) 5 (d) 81
38. 38.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 9)
39. 39.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 7) (c) (x − 3)(x − 5) (d) (x + 1)(x + 15)
40. 40.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 14 (d) 15

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