Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 143 · 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 81²? (a) 6561 (b) 6401 (c) 6571 (d) 6241
2. 2.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 8075 (b) 9025 (c) 7225 (d) 7235
3. 3.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 113 (c) 169 (d) 337
4. 4.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 1 and 12 (c) 2 and 18 (d) 4 and 9
5. 5.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 16 (c) 10 (d) 4
6. 6.What is the product of the roots of x² − 11x + 28 = 0? (a) 11 (b) 27 (c) 28 (d) 29
7. 7.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 81 (c) 61 (d) 41
8. 8.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 4634 (c) 5184 (d) 4624
9. 9.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 11 (c) 30 (d) 10
10. 10.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 916 (c) 1156 (d) 676
11. 11.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 47 (c) 14 (d) 48
12. 12.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 56 (c) 1 and 14 (d) 2 and 28
13. 13.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 19 (c) 11 (d) 17
14. 14.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 6 (c) 324 (d) 36
15. 15.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 36 (c) 31 (d) 26
16. 16.What are the roots of x² − 11x + 18 = 0? (a) 2 and 9 (b) 1 and 18 (c) 1 and 10 (d) 3 and 6
17. 17.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
18. 18.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 73 (d) 17
19. 19.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 399 (c) 441 (d) 371
20. 20.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 1 and 8 (c) 2 and 5 (d) 1 and 5
21. 21.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 4 (d) 5
22. 22.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 63 (d) 64
23. 23.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
24. 24.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 65 (d) 177
25. 25.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 4 (d) 400
26. 26.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
27. 27.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
28. 28.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
29. 29.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 3 and 21 (d) 7 and 9
30. 30.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 10609 (c) 9419 (d) 9409
31. 31.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 901 (c) 841 (d) 971
32. 32.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 9 (c) 149 (d) 3
33. 33.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 3)(x + 5)
34. 34.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7579 (c) 7569 (d) 8091
35. 35.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 8281 (c) 7931 (d) 8099
36. 36.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 7 (c) 8 (d) 12
37. 37.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 9 (c) 17 (d) 3
38. 38.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 201 (c) 146 (d) 366
39. 39.What are the roots of x² − 7x + 12 = 0? (a) 1 and 12 (b) 1 and 6 (c) 2 and 6 (d) 3 and 4
40. 40.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 299 (c) 391 (d) 529

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