Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 179 · 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) 5625 (c) 7225 (d) 6425
2. 2.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1681 (d) 1691
3. 3.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 1 and 10 (c) 5 and 6 (d) 2 and 15
4. 4.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
5. 5.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 10001 (d) 9801
6. 6.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 1)(x + 12)
7. 7.What is the product of the roots of x² − 10x + 16 = 0? (a) 16 (b) 10 (c) 15 (d) 17
8. 8.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 7 and 8 (d) 1 and 14
9. 9.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 196 (c) 2 (d) 4
10. 10.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
11. 11.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3721 (c) 3599 (d) 3491
12. 12.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
13. 13.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 10 (c) 21 (d) 20
14. 14.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 1 (c) 271 (d) 181
15. 15.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 16 (d) 1 and 72
16. 16.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 4925 (c) 4225 (d) 5635
17. 17.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 12) (c) (x − 2)(x − 6) (d) (x + 1)(x + 7)
18. 18.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 169 (c) 289 (d) 49
19. 19.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 361 (c) 529 (d) 193
20. 20.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4634 (c) 4896 (d) 4624
21. 21.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2611 (c) 2501 (d) 2601
22. 22.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
23. 23.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 52 (c) 20 (d) 36
24. 24.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 16 (c) 15 (d) 54
25. 25.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 71 (d) 17
26. 26.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 74 (c) 144 (d) 214
27. 27.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 384 (c) 576 (d) 266
28. 28.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9975 (c) 11025 (d) 9025
29. 29.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x + 4)(x + 9) (d) (x − 4)(x − 9)
30. 30.What are the roots of x² − 7x + 12 = 0? (a) 1 and 12 (b) 2 and 6 (c) 3 and 4 (d) 1 and 6
31. 31.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 15 (d) 14
32. 32.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 117 (c) 225 (d) 333
33. 33.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 2 and 6 (c) 3 and 4 (d) 1 and 12
34. 34.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 13 (c) 11 (d) 32
35. 35.If a + b = 11 and ab = 10, what is a² + b²? (a) 101 (b) 111 (c) 121 (d) 141
36. 36.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 12 (c) 11 (d) 24
37. 37.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 15 (c) 7 and 9 (d) 1 and 63
38. 38.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 97 (c) 241 (d) 133
39. 39.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 5)(x + 7) (c) (x + 1)(x + 11) (d) (x − 5)(x − 7)
40. 40.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2304 (b) 2704 (c) 2314 (d) 2496

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