Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 650 · 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) 10201 (b) 10001 (c) 9801 (d) 10211
2. 2.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 149 (c) 289 (d) 3
3. 3.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 1)(x + 12) (d) (x + 5)(x + 8)
4. 4.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1235 (c) 625 (d) 1225
5. 5.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2314 (c) 2304 (d) 2496
6. 6.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 4 (c) 10 (d) 2
7. 7.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
8. 8.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 484 (c) 494 (d) 404
9. 9.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 28)
10. 10.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 181 (c) 313 (d) 137
11. 11.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 36 (c) 34 (d) 35
12. 12.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 3 and 5 (c) 1 and 15 (d) 2 and 6
13. 13.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 2)(x + 6) (c) (x − 2)(x − 6) (d) (x + 1)(x + 12)
14. 14.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 72 (d) 1 and 16
15. 15.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 1)(x + 7) (d) (x + 3)(x + 5)
16. 16.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 361 (c) 185 (d) 9
17. 17.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x − 3)(x − 8) (c) (x + 1)(x + 24) (d) (x + 1)(x + 10)
18. 18.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 18 (c) 10 (d) 11
19. 19.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 7225 (c) 9025 (d) 8125
20. 20.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 333 (c) 441 (d) 225
21. 21.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 15 (c) 3 and 21 (d) 1 and 63
22. 22.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 74 (c) 2 (d) 144
23. 23.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 16 (c) 170 (d) 324
24. 24.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 23 (c) 25 (d) 24
25. 25.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
26. 26.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 17 (c) 11 (d) 18
27. 27.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 11)
28. 28.If a + b = 9 and ab = 8, what is (a − b)²? (a) 81 (b) 65 (c) 49 (d) 7
29. 29.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 31 (d) 29
30. 30.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
31. 31.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 396 (c) 484 (d) 324
32. 32.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 17 (d) 15
33. 33.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 11 (c) 30 (d) 10
34. 34.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 1 and 10 (c) 4 and 7 (d) 1 and 28
35. 35.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6724 (c) 6094 (d) 6396
36. 36.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 49 (c) 39 (d) 69
37. 37.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
38. 38.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 4225 (c) 3035 (d) 3025
39. 39.If a + b = 10 and ab = 9, what is (a − b)²? (a) 8 (b) 82 (c) 100 (d) 64
40. 40.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 12 (c) 13 (d) 14

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