Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 310 · 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 − 8x + 12 = 0? (a) 12 (b) 11 (c) 8 (d) 13
2. 2.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
3. 3.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 8474 (c) 8464 (d) 7744
4. 4.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
5. 5.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 11 (c) 12 (d) 32
6. 6.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
7. 7.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 1)(x + 4) (d) (x + 2)(x + 3)
8. 8.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 25 (c) 17 (d) 21
9. 9.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 113 (c) 169 (d) 225
10. 10.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 113 (c) 169 (d) 225
11. 11.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 8836 (c) 7396 (d) 8084
12. 12.If a + b = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 169 (c) 121 (d) 145
13. 13.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
14. 14.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 400 (c) 4 (d) 208
15. 15.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 65 (c) 121 (d) 9
16. 16.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1616 (d) 1936
17. 17.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 9 (c) 8 (d) 10
18. 18.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 169 (c) 9 (d) 3
19. 19.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1774 (c) 1444 (d) 1604
20. 20.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 184 (c) 144 (d) 104
21. 21.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3249 (b) 3969 (c) 3591 (d) 3259
22. 22.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
23. 23.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 6391 (c) 5939 (d) 6889
24. 24.If a + b = 15 and ab = 44, what is a² + b²? (a) 313 (b) 225 (c) 137 (d) 181
25. 25.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x − 6)(x − 9)
26. 26.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 46 (c) 14 (d) 45
27. 27.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 7754 (c) 8464 (d) 7744
28. 28.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 5625 (d) 4875
29. 29.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10404 (c) 10414 (d) 9604
30. 30.If a + b = 12 and ab = 32, what is (a − b)²? (a) 16 (b) 144 (c) 80 (d) 4
31. 31.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
32. 32.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 9)
33. 33.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 18 (c) 9 (d) 17
34. 34.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
35. 35.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
36. 36.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 2 and 15 (c) 1 and 30 (d) 1 and 10
37. 37.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8291 (c) 8281 (d) 7921
38. 38.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1599 (c) 1521 (d) 1681
39. 39.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 127 (c) 253 (d) 169
40. 40.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 217 (d) 145

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