Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 882 · 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 sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 15 (c) 13 (d) 48
2. 2.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 9409 (c) 10619 (d) 10009
3. 3.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 5 and 6 (c) 2 and 15 (d) 1 and 10
4. 4.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 19 (c) 13 (d) 25
5. 5.If a + b = 12 and ab = 27, what is (a − b)²? (a) 36 (b) 90 (c) 6 (d) 144
6. 6.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 21 (d) 19
7. 7.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 15 (d) 14
8. 8.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 85 (c) 169 (d) 1
9. 9.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5051 (c) 5041 (d) 4901
10. 10.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 121 (c) 81 (d) 9
11. 11.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3481 (c) 3601 (d) 3721
12. 12.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 39 (c) 13 (d) 40
13. 13.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
14. 14.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3025 (c) 3575 (d) 4225
15. 15.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 2 and 24 (c) 6 and 8 (d) 1 and 13
16. 16.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
17. 17.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 1 and 12 (c) 5 and 8 (d) 1 and 40
18. 18.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 11 (c) 35 (d) 13
19. 19.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x − 4)(x − 9)
20. 20.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)
21. 21.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)
22. 22.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1681 (c) 1521 (d) 1601
23. 23.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)
24. 24.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 6)
25. 25.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 909 (c) 1089 (d) 1099
26. 26.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3979 (c) 3969 (d) 3249
27. 27.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 586 (d) 576
28. 28.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 598 (c) 301 (d) 202
29. 29.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 2 and 28 (c) 1 and 56 (d) 1 and 14
30. 30.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 361 (c) 529 (d) 193
31. 31.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1604 (c) 1774 (d) 1444
32. 32.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
33. 33.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
34. 34.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 3)(x + 5) (c) (x − 3)(x − 5) (d) (x + 1)(x + 15)
35. 35.If a + b = 9 and ab = 14, what is a² + b²? (a) 53 (b) 67 (c) 81 (d) 109
36. 36.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 2)(x + 5) (c) (x + 1)(x + 6) (d) (x + 1)(x + 10)
37. 37.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
38. 38.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 10001 (d) 9801
39. 39.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 244 (c) 484 (d) 164
40. 40.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 273 (c) 185 (d) 361

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