Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 116 · 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.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 56 (d) 57
2. 2.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
4. 4.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 1 and 72 (d) 8 and 9
5. 5.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 61 (c) 41 (d) 81
6. 6.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3721 (c) 3481 (d) 3601
7. 7.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 196 (c) 156 (d) 116
8. 8.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 193 (c) 361 (d) 529
9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
10. 10.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 36 (c) 256 (d) 6
11. 11.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9409 (c) 9419 (d) 9991
12. 12.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
13. 13.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 64 (c) 62 (d) 16
14. 14.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 16 (c) 54 (d) 14
15. 15.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 185 (c) 3 (d) 9
16. 16.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 1 and 8 (c) 1 and 5 (d) 2 and 5
17. 17.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 9801 (c) 10211 (d) 10201
18. 18.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 9)
19. 19.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 121 (c) 141 (d) 101
20. 20.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6084 (c) 6396 (d) 6094
21. 21.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1691 (d) 1681
22. 22.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 17 (c) 15 (d) 16
23. 23.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
24. 24.If a + b = 6 and ab = 8, what is (a − b)²? (a) 36 (b) 4 (c) 20 (d) 2
25. 25.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
26. 26.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 3 (c) 9 (d) 29
27. 27.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 73 (d) 71
28. 28.If a + b = 14 and ab = 33, what is (a − b)²? (a) 64 (b) 8 (c) 130 (d) 196
29. 29.If a + b = 13 and ab = 36, what is (a − b)²? (a) 169 (b) 97 (c) 5 (d) 25
30. 30.If a + b = 11 and ab = 10, what is (a − b)²? (a) 9 (b) 121 (c) 81 (d) 101
31. 31.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 361 (c) 273 (d) 185
32. 32.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 1 and 12 (c) 2 and 20 (d) 1 and 40
33. 33.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4771 (d) 4761
34. 34.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 12 (c) 32 (d) 11
35. 35.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 324 (c) 478 (d) 247
36. 36.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 4 (c) 2 (d) 34
37. 37.What are the roots of x² − 14x + 45 = 0? (a) 5 and 9 (b) 3 and 15 (c) 1 and 13 (d) 1 and 45
38. 38.What is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 14 (c) 13 (d) 9
39. 39.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 1)(x + 40)
40. 40.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 3025 (c) 2025 (d) 2035

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