Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 706 · 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 + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 54) (c) (x + 1)(x + 14) (d) (x − 6)(x − 9)
2. 2.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)
3. 3.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 4)
4. 4.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 169 (c) 9 (d) 89
5. 5.If a + b = 13 and ab = 22, what is a² + b²? (a) 147 (b) 125 (c) 213 (d) 169
6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
7. 7.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 537 (c) 361 (d) 273
8. 8.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 125 (c) 81 (d) 169
9. 9.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 9 (c) 11 (d) 10
10. 10.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 170 (c) 324 (d) 247
11. 11.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 361 (c) 181 (d) 271
12. 12.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
13. 13.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
14. 14.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2035 (c) 2475 (d) 2025
15. 15.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1599 (c) 1531 (d) 1681
16. 16.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 2 and 9 (c) 3 and 6 (d) 1 and 10
17. 17.What is the product of the roots of x² − 11x + 24 = 0? (a) 24 (b) 11 (c) 23 (d) 25
18. 18.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 37 (c) 25 (d) 49
19. 19.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 113 (c) 1 (d) 225
20. 20.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 17 (c) 63 (d) 16
21. 21.If a + b = 14 and ab = 40, what is (a − b)²? (a) 36 (b) 6 (c) 116 (d) 196
22. 22.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 400 (c) 304 (d) 592
23. 23.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 371 (c) 441 (d) 361
24. 24.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2401 (c) 2501 (d) 2611
25. 25.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 73 (c) 65 (d) 81
26. 26.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 109 (c) 214 (d) 74
27. 27.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3591 (c) 3249 (d) 3259
28. 28.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 11 (c) 10 (d) 9
29. 29.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2509 (c) 2209 (d) 2809
30. 30.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 252 (c) 180 (d) 468
31. 31.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 324 (c) 170 (d) 4
32. 32.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 11 (c) 12 (d) 13
33. 33.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 2 and 12 (c) 4 and 6 (d) 1 and 24
34. 34.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4225 (c) 4235 (d) 4875
35. 35.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 2 (d) 4
36. 36.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
37. 37.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 494 (c) 404 (d) 484
38. 38.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 15 (d) 56
39. 39.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 256 (c) 366 (d) 146
40. 40.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7744 (c) 8096 (d) 7754

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