Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 267 · 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 + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x − 3)(x − 9)
2. 2.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 5625 (d) 4875
3. 3.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 1156 (c) 676 (d) 686
4. 4.If a + b = 16 and ab = 55, what is (a − b)²? (a) 36 (b) 6 (c) 256 (d) 146
5. 5.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x − 5)(x − 8)
6. 6.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
7. 7.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 2 (c) 16 (d) 10
8. 8.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 324 (c) 180 (d) 252
9. 9.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
10. 10.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 304 (c) 400 (d) 592
11. 11.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)
12. 12.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 25 (c) 11 (d) 24
13. 13.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
14. 14.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
15. 15.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 26 (c) 4 (d) 16
16. 16.What are the roots of x² − 10x + 24 = 0? (a) 2 and 12 (b) 4 and 6 (c) 1 and 9 (d) 1 and 24
17. 17.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 136 (c) 196 (d) 256
18. 18.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 1)(x + 4)
19. 19.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 11025 (c) 11035 (d) 9025
20. 20.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 297 (c) 189 (d) 153
21. 21.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 11025 (c) 9035 (d) 9025
22. 22.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 325 (c) 175 (d) 225
23. 23.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 56 (d) 57
24. 24.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 30 (c) 5 and 6 (d) 1 and 10
25. 25.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 225 (c) 117 (d) 333
26. 26.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 219 (c) 149 (d) 289
27. 27.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 72 (c) 1 and 16 (d) 2 and 36
28. 28.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)
29. 29.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2611 (c) 2401 (d) 2601
30. 30.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
31. 31.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 37 (c) 19 (d) 13
32. 32.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2304 (c) 2504 (d) 2714
33. 33.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
34. 34.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)
35. 35.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 494 (c) 484 (d) 404
36. 36.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 73 (c) 169 (d) 97
37. 37.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 15 (d) 13
38. 38.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3491 (c) 3481 (d) 3599
39. 39.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 10 (c) 12 (d) 11
40. 40.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 127 (c) 85 (d) 1

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