Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 468 · 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.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 2 and 8 (c) 3 and 7 (d) 1 and 21
2. 2.If a + b = 14 and ab = 40, what is a² + b²? (a) 156 (b) 196 (c) 116 (d) 276
3. 3.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 12 (c) 13 (d) 14
4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
5. 5.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6241 (c) 6561 (d) 6399
6. 6.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 2 and 27 (c) 6 and 9 (d) 1 and 54
7. 7.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3969 (c) 3609 (d) 3249
8. 8.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1599 (c) 1521 (d) 1531
9. 9.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 8075 (b) 7235 (c) 9025 (d) 7225
10. 10.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5194 (c) 4624 (d) 5184
11. 11.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 396 (c) 334 (d) 484
12. 12.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 48 (c) 13 (d) 15
13. 13.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 223 (c) 289 (d) 157
14. 14.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 4 (c) 2 (d) 74
15. 15.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
16. 16.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)
17. 17.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 73 (d) 17
18. 18.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 133 (c) 97 (d) 241
19. 19.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 169 (c) 89 (d) 9
20. 20.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 2 and 36 (d) 8 and 9
21. 21.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 1)(x + 14) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
22. 22.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
23. 23.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 10 (c) 18 (d) 9
24. 24.If a + b = 12 and ab = 32, what is a² + b²? (a) 80 (b) 144 (c) 208 (d) 112
25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
26. 26.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 196 (c) 292 (d) 100
27. 27.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 42 (c) 13 (d) 43
28. 28.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 125 (c) 81 (d) 169
29. 29.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
30. 30.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 48 (c) 49 (d) 14
31. 31.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 169 (c) 113 (d) 225
32. 32.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1444 (c) 1454 (d) 1764
33. 33.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 277 (c) 529 (d) 193
34. 34.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 10004 (c) 9604 (d) 10404
35. 35.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 371 (c) 441 (d) 399
36. 36.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 3 (c) 289 (d) 149
37. 37.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3481 (c) 3601 (d) 3731
38. 38.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
39. 39.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
40. 40.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2304 (c) 2704 (d) 2714

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