Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 848 · 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.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
2. 2.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1774 (c) 1604 (d) 1764
3. 3.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 10 (c) 9 (d) 8
4. 4.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
5. 5.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 1 and 72 (d) 2 and 36
6. 6.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 7744 (c) 8104 (d) 8474
7. 7.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 16 (c) 2 (d) 4
8. 8.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 9 (c) 7 (d) 8
9. 9.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1936 (c) 1306 (d) 1584
10. 10.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 1024 (c) 794 (d) 896
11. 11.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 1)(x + 12)
12. 12.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 8109 (c) 8649 (d) 7569
13. 13.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 13 (c) 1 (d) 19
14. 14.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
15. 15.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 541 (d) 271
16. 16.What is the product of the roots of x² − 9x + 20 = 0? (a) 20 (b) 21 (c) 19 (d) 9
17. 17.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 324 (c) 4 (d) 16
18. 18.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 366 (c) 256 (d) 201
19. 19.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3844 (c) 3596 (d) 3364
20. 20.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 7 (c) 9 (d) 1
21. 21.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 416 (c) 256 (d) 586
22. 22.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6404 (c) 6084 (d) 6724
23. 23.What are the roots of x² − 14x + 45 = 0? (a) 5 and 9 (b) 3 and 15 (c) 1 and 13 (d) 1 and 45
24. 24.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 30 (c) 11 (d) 29
25. 25.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 25 (c) 21 (d) 17
26. 26.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6561 (c) 6571 (d) 6241
27. 27.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 16 (c) 58 (d) 100
28. 28.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2411 (c) 2499 (d) 2401
29. 29.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 4 (c) 52 (d) 2
30. 30.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 160 (c) 8 (d) 64
31. 31.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3025 (c) 2025 (d) 3035
32. 32.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 13 (c) 11 (d) 12
33. 33.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 361 (c) 441 (d) 371
34. 34.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2126 (c) 2484 (d) 2916
35. 35.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 9999 (c) 10201 (d) 9801
36. 36.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)
37. 37.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 74 (c) 214 (d) 144
38. 38.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
39. 39.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 1 (c) 225 (d) 113
40. 40.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 266 (c) 384 (d) 256

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