Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 597 · 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 − 16x + 63 = 0? (a) 63 (b) 15 (c) 17 (d) 16
2. 2.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 441 (c) 333 (d) 657
3. 3.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 7 and 9 (c) 1 and 63 (d) 1 and 15
4. 4.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4896 (c) 4624 (d) 5184
5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
6. 6.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 62 (c) 64 (d) 16
7. 7.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 29 (c) 28 (d) 11
8. 8.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)
9. 9.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1521 (c) 1691 (d) 1601
10. 10.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 289 (c) 529 (d) 409
11. 11.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
12. 12.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 18 (c) 19 (d) 17
13. 13.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 10404 (c) 9614 (d) 9996
14. 14.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 484 (c) 334 (d) 324
15. 15.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 25 (c) 157 (d) 5
16. 16.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
17. 17.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 6 (c) 116 (d) 36
18. 18.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
19. 19.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4366 (c) 4884 (d) 4356
20. 20.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 148 (c) 52 (d) 100
21. 21.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 17 (c) 16 (d) 10
22. 22.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
23. 23.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 19 (c) 21 (d) 20
24. 24.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
25. 25.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 494 (c) 484 (d) 404
26. 26.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 313 (c) 181 (d) 137
27. 27.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1369 (c) 1609 (d) 1849
28. 28.If a + b = 15 and ab = 36, what is a² + b²? (a) 189 (b) 297 (c) 153 (d) 225
29. 29.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 7) (c) (x + 1)(x + 12) (d) (x − 2)(x − 6)
30. 30.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9216 (c) 9984 (d) 10816
31. 31.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 12 (c) 36 (d) 35
32. 32.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 10 (c) 11 (d) 18
33. 33.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 10)
34. 34.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 12 (d) 10
35. 35.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 22 (c) 20 (d) 21
36. 36.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x + 1)(x + 16) (d) (x − 2)(x − 8)
37. 37.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 54 (c) 2 and 27 (d) 1 and 14
38. 38.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 100 (c) 68 (d) 132
39. 39.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6084 (c) 6094 (d) 6396
40. 40.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 4)(x + 7)

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