Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 357 · 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 + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
2. 2.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x + 4)(x + 7) (d) (x − 4)(x − 7)
3. 3.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 9409 (c) 9419 (d) 10609
4. 4.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 4761 (c) 5041 (d) 4899
5. 5.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 79 (c) 142 (d) 58
6. 6.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)
7. 7.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 11 (c) 29 (d) 30
8. 8.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 289 (c) 429 (d) 149
9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
10. 10.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 333 (c) 441 (d) 657
11. 11.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 125 (c) 225 (d) 175
12. 12.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
13. 13.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
14. 14.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 25 (c) 49 (d) 73
15. 15.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 4225 (c) 5635 (d) 5625
16. 16.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 27 (c) 11 (d) 13
17. 17.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 3625 (d) 3025
18. 18.What is the product of the roots of x² − 8x + 15 = 0? (a) 14 (b) 8 (c) 15 (d) 16
19. 19.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10414 (c) 10404 (d) 9604
20. 20.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 4 (c) 106 (d) 196
21. 21.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 45 (c) 9 (d) 81
22. 22.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 8464 (c) 7744 (d) 8474
23. 23.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 71 (d) 72
24. 24.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
25. 25.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2611 (c) 2401 (d) 2601
26. 26.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 196 (c) 4 (d) 100
27. 27.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 3 and 15 (c) 5 and 9 (d) 1 and 13
28. 28.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 133 (c) 144 (d) 122
29. 29.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 7225 (c) 5625 (d) 6375
30. 30.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 39 (c) 29 (d) 49
31. 31.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 3136 (c) 3146 (d) 4096
32. 32.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
33. 33.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 2 and 20 (c) 1 and 12 (d) 5 and 8
34. 34.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2809 (c) 2509 (d) 2209
35. 35.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 1 and 5 (c) 2 and 5 (d) 2 and 4
36. 36.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)
37. 37.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 64 (c) 34 (d) 49
38. 38.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 875 (c) 1225 (d) 625
39. 39.If a + b = 12 and ab = 27, what is (a − b)²? (a) 36 (b) 144 (c) 6 (d) 90
40. 40.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 201 (c) 366 (d) 146

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