Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 358 · 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.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 52 (c) 100 (d) 76
2. 2.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1306 (c) 1296 (d) 1584
3. 3.If a + b = 13 and ab = 36, what is a² + b²? (a) 97 (b) 169 (c) 241 (d) 133
4. 4.If a + b = 13 and ab = 12, what is (a − b)²? (a) 121 (b) 169 (c) 145 (d) 11
5. 5.What are the roots of x² − 9x + 20 = 0? (a) 4 and 5 (b) 1 and 8 (c) 1 and 20 (d) 2 and 10
6. 6.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
7. 7.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 72 (d) 17
8. 8.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
9. 9.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
10. 10.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1849 (c) 1609 (d) 1369
11. 11.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 6 (c) 7 (d) 8
12. 12.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 16 (c) 17 (d) 15
13. 13.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 5 (c) 25 (d) 169
14. 14.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8101 (c) 7921 (d) 8281
15. 15.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 382 (c) 256 (d) 130
16. 16.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
17. 17.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 121 (c) 41 (d) 81
18. 18.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
19. 19.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 16 (d) 15
20. 20.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 36 (c) 28 (d) 20
21. 21.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 1 and 18 (c) 2 and 9 (d) 1 and 10
22. 22.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)
23. 23.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 2 and 24 (c) 1 and 13 (d) 1 and 48
24. 24.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 441 (c) 371 (d) 399
25. 25.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 8091 (c) 7569 (d) 7579
26. 26.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x + 2)(x + 9)
27. 27.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1681 (c) 1599 (d) 1531
28. 28.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5625 (c) 5635 (d) 4225
29. 29.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 14)
30. 30.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 9 (c) 225 (d) 3
31. 31.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 13 (c) 43 (d) 42
32. 32.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 2025 (c) 1575 (d) 1225
33. 33.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
34. 34.What is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 9 (c) 13 (d) 14
35. 35.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 441 (c) 225 (d) 333
36. 36.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3596 (c) 3364 (d) 3374
37. 37.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 169 (c) 157 (d) 193
38. 38.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8096 (c) 7744 (d) 8464
39. 39.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
40. 40.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 112 (c) 144 (d) 80

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