Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 146 · 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.If a + b = 12 and ab = 32, what is a² + b²? (a) 144 (b) 208 (c) 80 (d) 112
2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
3. 3.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 9801 (c) 10201 (d) 9811
4. 4.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 39 (c) 49 (d) 29
5. 5.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 36 (c) 35 (d) 12
6. 6.If a + b = 7 and ab = 10, what is (a − b)²? (a) 3 (b) 49 (c) 9 (d) 29
7. 7.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 56 (d) 15
8. 8.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
9. 9.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)
10. 10.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)
11. 11.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 4489 (c) 5329 (d) 5339
12. 12.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6409 (c) 6899 (d) 6889
13. 13.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 1099 (c) 729 (d) 909
14. 14.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 4096 (c) 3146 (d) 3136
15. 15.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 90 (c) 144 (d) 198
16. 16.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 537 (c) 185 (d) 361
17. 17.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
18. 18.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 193 (c) 361 (d) 277
19. 19.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 6375 (c) 5625 (d) 7225
20. 20.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 247 (c) 324 (d) 478
21. 21.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 5625 (c) 7225 (d) 7235
22. 22.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
23. 23.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 598 (c) 301 (d) 202
24. 24.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
25. 25.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4624 (b) 4634 (c) 4896 (d) 5184
26. 26.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
27. 27.What is the product of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 41 (d) 39
28. 28.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 73 (c) 5 (d) 25
29. 29.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 7 and 8 (d) 1 and 56
30. 30.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 5)(x + 8) (c) (x + 1)(x + 12) (d) (x + 1)(x + 40)
31. 31.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
32. 32.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
33. 33.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 19 (c) 25 (d) 1
34. 34.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 7 (c) 6 (d) 12
35. 35.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
36. 36.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 130 (c) 382 (d) 256
37. 37.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 8 (d) 5
38. 38.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 925 (c) 1235 (d) 625
39. 39.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
40. 40.If a + b = 6 and ab = 5, what is (a − b)²? (a) 4 (b) 26 (c) 16 (d) 36

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