Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 458 · 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 is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 14 (c) 49 (d) 48
2. 2.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
3. 3.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 185 (c) 9 (d) 3
4. 4.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 169 (c) 225 (d) 1
5. 5.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 884 (c) 1156 (d) 676
6. 6.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4761 (c) 4771 (d) 4899
7. 7.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 25 (c) 3 (d) 9
8. 8.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 9 (c) 89 (d) 169
9. 9.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 8281 (c) 7921 (d) 7931
10. 10.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
11. 11.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 361 (c) 441 (d) 371
12. 12.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 1 and 63 (c) 7 and 9 (d) 3 and 21
13. 13.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 49 (c) 37 (d) 25
14. 14.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 100 (c) 2 (d) 52
15. 15.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5329 (b) 4489 (c) 4909 (d) 5339
16. 16.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 21 (c) 20 (d) 9
17. 17.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 148 (c) 196 (d) 100
18. 18.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 7 (c) 12 (d) 6
19. 19.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 4 (c) 196 (d) 106
20. 20.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 1 and 16 (d) 8 and 9
21. 21.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 26 (c) 12 (d) 28
22. 22.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 63 (d) 16
23. 23.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 1089 (c) 739 (d) 729
24. 24.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 3 (c) 9 (d) 289
25. 25.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
26. 26.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 441 (c) 451 (d) 361
27. 27.What are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 1 and 12 (c) 1 and 36 (d) 4 and 9
28. 28.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 5184 (c) 4624 (d) 4896
29. 29.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2916 (c) 2116 (d) 2926
30. 30.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 9 (c) 225 (d) 3
31. 31.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 3)(x + 9) (d) (x + 1)(x + 11)
32. 32.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 7569 (c) 8649 (d) 8091
33. 33.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
34. 34.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
35. 35.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 12 (c) 13 (d) 14
36. 36.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 1225 (c) 635 (d) 875
37. 37.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 14 (c) 13 (d) 42
38. 38.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 17 (d) 16
39. 39.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 79 (c) 142 (d) 100
40. 40.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x + 1)(x + 16) (d) (x − 2)(x − 8)

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