Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 337 · 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.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10001 (c) 10211 (d) 10201
2. 2.If a + b = 13 and ab = 22, what is a² + b²? (a) 213 (b) 147 (c) 169 (d) 125
3. 3.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
4. 4.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 13 (c) 11 (d) 12
5. 5.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)
6. 6.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)
7. 7.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
8. 8.If a + b = 13 and ab = 40, what is (a − b)²? (a) 9 (b) 89 (c) 169 (d) 3
9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
10. 10.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 286 (c) 196 (d) 151
11. 11.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
12. 12.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 739 (c) 891 (d) 729
13. 13.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 16 (c) 10 (d) 9
14. 14.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 3 (c) 9 (d) 361
15. 15.If a + b = 12 and ab = 20, what is (a − b)²? (a) 64 (b) 8 (c) 104 (d) 144
16. 16.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 36 (c) 324 (d) 180
17. 17.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5486 (d) 5476
18. 18.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7931 (b) 8281 (c) 8099 (d) 7921
19. 19.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 16 (c) 15 (d) 14
20. 20.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 333 (c) 117 (d) 171
21. 21.If a + b = 3 and ab = 2, what is a² + b²? (a) 7 (b) 13 (c) 9 (d) 5
22. 22.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 144 (c) 4 (d) 74
23. 23.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 175 (c) 225 (d) 325
24. 24.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 324 (c) 404 (d) 484
25. 25.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)
26. 26.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 2 and 12 (c) 1 and 24 (d) 3 and 8
27. 27.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1774 (c) 1444 (d) 1604
28. 28.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 19 (c) 13 (d) 37
29. 29.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3601 (c) 3481 (d) 3721
30. 30.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x − 2)(x − 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
31. 31.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
32. 32.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3575 (c) 3025 (d) 4225
33. 33.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 6 (c) 7 (d) 10
34. 34.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
35. 35.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3374 (c) 3364 (d) 3596
36. 36.If a + b = 14 and ab = 24, what is (a − b)²? (a) 100 (b) 10 (c) 196 (d) 148
37. 37.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x − 6)(x − 9)
38. 38.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9984 (c) 9226 (d) 9216
39. 39.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)
40. 40.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 1 and 63 (c) 3 and 21 (d) 7 and 9

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