Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 284 · 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 are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 2 and 6 (c) 3 and 5 (d) 1 and 7
2. 2.If a + b = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 180 (c) 324 (d) 6
3. 3.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x + 1)(x + 10)
4. 4.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
5. 5.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2035 (c) 2025 (d) 2475
6. 6.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 13 (c) 45 (d) 14
7. 7.If a + b = 14 and ab = 40, what is (a − b)²? (a) 6 (b) 196 (c) 36 (d) 116
8. 8.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
9. 9.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 61 (c) 81 (d) 41
10. 10.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 208 (c) 160 (d) 352
11. 11.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1625 (c) 2025 (d) 1225
12. 12.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)
13. 13.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2314 (c) 2304 (d) 2704
14. 14.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4235 (c) 4225 (d) 4875
15. 15.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1444 (c) 1454 (d) 1764
16. 16.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 4 (c) 256 (d) 130
17. 17.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 11 (c) 12 (d) 35
18. 18.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
19. 19.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 72 (d) 16
20. 20.If a + b = 7 and ab = 10, what is (a − b)²? (a) 3 (b) 9 (c) 49 (d) 29
21. 21.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 40)
22. 22.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
23. 23.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 1 and 40 (c) 2 and 20 (d) 5 and 8
24. 24.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 851 (c) 961 (d) 841
25. 25.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5625 (b) 6375 (c) 7225 (d) 5635
26. 26.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 145 (d) 217
27. 27.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 10 (c) 2 and 15 (d) 1 and 30
28. 28.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7931 (b) 8099 (c) 7921 (d) 8281
29. 29.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 9 (c) 5 (d) 1
30. 30.If a + b = 13 and ab = 22, what is a² + b²? (a) 147 (b) 125 (c) 169 (d) 213
31. 31.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 169 (c) 225 (d) 113
32. 32.If a + b = 16 and ab = 48, what is (a − b)²? (a) 160 (b) 256 (c) 64 (d) 8
33. 33.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 5476 (c) 4366 (d) 4884
34. 34.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)
35. 35.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16
36. 36.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 8) (c) (x + 1)(x + 14) (d) (x − 2)(x − 7)
37. 37.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 57 (d) 56
38. 38.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 181 (d) 1
39. 39.If a + b = 12 and ab = 11, what is (a − b)²? (a) 122 (b) 100 (c) 10 (d) 144
40. 40.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9604 (c) 9996 (d) 9614

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