Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 617 · 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 23²? (a) 409 (b) 529 (c) 289 (d) 539
2. 2.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 441 (c) 361 (d) 399
3. 3.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 11 (c) 13 (d) 27
4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
5. 5.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
6. 6.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 43 (c) 61 (d) 37
7. 7.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 7744 (c) 8474 (d) 8464
8. 8.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 1 (c) 61 (d) 81
9. 9.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 289 (c) 219 (d) 149
10. 10.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 2 and 5 (c) 1 and 6 (d) 1 and 10
11. 11.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 16 (c) 400 (d) 208
12. 12.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 10) (c) (x + 1)(x + 6) (d) (x − 2)(x − 5)
13. 13.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 14 (c) 8 (d) 15
14. 14.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 1 and 12 (c) 5 and 8 (d) 1 and 40
15. 15.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 375 (c) 235 (d) 625
16. 16.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 14 (d) 15
17. 17.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 1 and 24 (c) 2 and 12 (d) 3 and 8
18. 18.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
19. 19.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
20. 20.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
21. 21.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
22. 22.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 2 and 10 (c) 1 and 27 (d) 1 and 11
23. 23.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2601 (c) 2401 (d) 2611
24. 24.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)
25. 25.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 8 (c) 64 (d) 104
26. 26.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 160 (c) 208 (d) 256
27. 27.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
28. 28.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 9 (c) 29 (d) 3
29. 29.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 5194 (c) 4904 (d) 5184
30. 30.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3604 (c) 3854 (d) 3364
31. 31.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 5625 (d) 4875
32. 32.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 225 (c) 441 (d) 3
33. 33.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 12 (c) 18 (d) 10
34. 34.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 21) (c) (x − 3)(x − 7) (d) (x + 1)(x + 9)
35. 35.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
36. 36.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 5 (c) 6 (d) 7
37. 37.What is the product of the roots of x² − 10x + 21 = 0? (a) 21 (b) 20 (c) 22 (d) 10
38. 38.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 30 (c) 11 (d) 31
39. 39.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6734 (c) 6084 (d) 6724
40. 40.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 9 (c) 3 (d) 289

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