Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 799 · 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 38²? (a) 1454 (b) 1764 (c) 1596 (d) 1444
2. 2.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 9 (c) 3 (d) 225
3. 3.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
4. 4.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 14 (c) 1 and 56 (d) 2 and 28
5. 5.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 130 (c) 163 (d) 262
6. 6.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5776 (c) 7056 (d) 5786
7. 7.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
8. 8.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
9. 9.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 625 (c) 1225 (d) 635
10. 10.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
11. 11.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7225 (b) 6425 (c) 7235 (d) 5625
12. 12.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 48 (c) 47 (d) 49
13. 13.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
14. 14.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 4)(x + 8) (c) (x + 1)(x + 32) (d) (x − 4)(x − 8)
15. 15.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 1 and 32 (c) 2 and 16 (d) 4 and 8
16. 16.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
17. 17.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6571 (c) 6241 (d) 6561
18. 18.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 494 (d) 484
19. 19.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 100 (c) 132 (d) 68
20. 20.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 73 (c) 121 (d) 97
21. 21.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)
22. 22.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
23. 23.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2819 (c) 2509 (d) 2809
24. 24.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
25. 25.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2401 (c) 2411 (d) 2601
26. 26.If a + b = 10 and ab = 9, what is (a − b)²? (a) 8 (b) 82 (c) 100 (d) 64
27. 27.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 676 (c) 1156 (d) 884
28. 28.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 15 (c) 14 (d) 13
29. 29.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 1 and 4 (c) 1 and 6 (d) 2 and 3
30. 30.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 891 (c) 729 (d) 1089
31. 31.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 3025 (c) 2025 (d) 2525
32. 32.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 127 (c) 1 (d) 85
33. 33.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)
34. 34.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 22 (c) 21 (d) 10
35. 35.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 289 (c) 433 (d) 217
36. 36.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 4 (c) 5 (d) 6
37. 37.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 7 (c) 12 (d) 13
38. 38.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)
39. 39.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)
40. 40.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 14) (d) (x + 1)(x + 8)

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