Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 782 · 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 54²? (a) 2926 (b) 2116 (c) 2516 (d) 2916
2. 2.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 301 (c) 202 (d) 598
3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
4. 4.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 121 (c) 101 (d) 111
5. 5.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
6. 6.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 175 (c) 325 (d) 225
7. 7.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)
8. 8.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 2 and 18 (c) 1 and 12 (d) 4 and 9
9. 9.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4499 (c) 4489 (d) 4891
10. 10.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5929 (c) 6391 (d) 5939
11. 11.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 2 and 21 (c) 6 and 7 (d) 1 and 12
12. 12.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 5 (c) 121 (d) 25
13. 13.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6241 (c) 6251 (d) 6399
14. 14.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 41 (c) 13 (d) 43
15. 15.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 225 (d) 9
16. 16.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1531 (c) 1599 (d) 1521
17. 17.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x − 4)(x − 7)
18. 18.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)
19. 19.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 144 (c) 109 (d) 74
20. 20.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 10009 (c) 10619 (d) 9409
21. 21.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 4 (c) 208 (d) 16
22. 22.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 30 (c) 11 (d) 29
23. 23.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 19 (c) 11 (d) 17
24. 24.If a + b = 7 and ab = 6, what is a² + b²? (a) 61 (b) 43 (c) 37 (d) 49
25. 25.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
26. 26.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 18) (c) (x − 3)(x − 6) (d) (x + 1)(x + 8)
27. 27.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 97 (c) 25 (d) 169
28. 28.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 72 (d) 16
29. 29.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x − 2)(x − 7)
30. 30.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 1 and 72 (d) 2 and 36
31. 31.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 324 (c) 494 (d) 404
32. 32.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 9 (c) 7 (d) 15
33. 33.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 25 (c) 23 (d) 24
34. 34.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 1225 (c) 925 (d) 625
35. 35.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 164 (c) 484 (d) 244
36. 36.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 16 (d) 14
37. 37.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 196 (c) 16 (d) 4
38. 38.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 7579 (c) 8649 (d) 7569
39. 39.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)
40. 40.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 324 (c) 478 (d) 247

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