Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 935 · 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 21²? (a) 401 (b) 361 (c) 451 (d) 441
2. 2.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
3. 3.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 32) (d) (x + 1)(x + 11)
4. 4.If a + b = 8 and ab = 15, what is (a − b)²? (a) 4 (b) 34 (c) 2 (d) 64
5. 5.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 164 (c) 2 (d) 324
6. 6.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1521 (c) 1531 (d) 1681
7. 7.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
8. 8.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
9. 9.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 117 (c) 9 (d) 3
10. 10.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x + 1)(x + 63)
11. 11.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 15 (d) 16
12. 12.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1296 (c) 1946 (d) 1936
13. 13.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 185 (c) 361 (d) 9
14. 14.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)
15. 15.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 289 (c) 49 (d) 7
16. 16.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)
17. 17.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 3035 (c) 2525 (d) 2025
18. 18.If a + b = 16 and ab = 48, what is a² + b²? (a) 160 (b) 256 (c) 208 (d) 352
19. 19.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 14 (c) 48 (d) 47
20. 20.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 36 (c) 6 (d) 116
21. 21.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
22. 22.What are the roots of x² − 14x + 45 = 0? (a) 3 and 15 (b) 1 and 45 (c) 1 and 13 (d) 5 and 9
23. 23.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 13 (c) 12 (d) 14
24. 24.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 3)(x + 7) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
25. 25.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6561 (c) 6241 (d) 6401
26. 26.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 4761 (c) 5041 (d) 5051
27. 27.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2704 (c) 2314 (d) 2304
28. 28.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 1 and 56 (d) 7 and 8
29. 29.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 5476 (c) 4356 (d) 5486
30. 30.If a + b = 14 and ab = 33, what is (a − b)²? (a) 8 (b) 130 (c) 64 (d) 196
31. 31.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2501 (c) 2401 (d) 2601
32. 32.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 9801 (c) 10201 (d) 9811
33. 33.If a + b = 13 and ab = 40, what is (a − b)²? (a) 9 (b) 169 (c) 89 (d) 3
34. 34.What is the product of the roots of x² − 12x + 32 = 0? (a) 33 (b) 12 (c) 31 (d) 32
35. 35.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 1)(x + 35)
36. 36.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 2 and 12 (c) 4 and 6 (d) 1 and 24
37. 37.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 12 (c) 11 (d) 10
38. 38.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
39. 39.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 100 (c) 148 (d) 52
40. 40.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 21 (c) 19 (d) 20

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