Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 835 · 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.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
2. 2.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 15 (c) 14 (d) 13
3. 3.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 85 (c) 169 (d) 127
4. 4.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
5. 5.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
6. 6.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 1)(x + 8)
7. 7.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 142 (c) 100 (d) 58
8. 8.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 478 (c) 170 (d) 324
9. 9.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2116 (c) 2926 (d) 2516
10. 10.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 484 (c) 324 (d) 396
11. 11.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 180 (c) 324 (d) 36
12. 12.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 3625 (d) 3025
13. 13.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2714 (c) 2304 (d) 2504
14. 14.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 925 (c) 1225 (d) 625
15. 15.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 225 (d) 337
16. 16.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1156 (c) 1166 (d) 676
17. 17.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
18. 18.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)
19. 19.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 399 (c) 441 (d) 361
20. 20.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 20 (c) 9 (d) 19
21. 21.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2611 (c) 2401 (d) 2501
22. 22.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 196 (c) 244 (d) 148
23. 23.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 1 and 11 (c) 3 and 9 (d) 2 and 10
24. 24.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 63 (d) 16
25. 25.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
26. 26.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 16 (c) 4 (d) 40
27. 27.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x + 1)(x + 12)
28. 28.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 16 (d) 14
29. 29.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 2 (c) 164 (d) 4
30. 30.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 10 (c) 22 (d) 13
31. 31.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 73 (d) 71
32. 32.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 9 (d) 225
33. 33.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 10 (c) 11 (d) 12
34. 34.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 9 (c) 8 (d) 14
35. 35.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 39 (c) 49 (d) 69
36. 36.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 235 (d) 625
37. 37.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 9811 (c) 10201 (d) 9801
38. 38.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 10 (c) 30 (d) 11
39. 39.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1599 (c) 1521 (d) 1531
40. 40.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 11)

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