Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 182 · 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 + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 1)(x + 10)
2. 2.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 361 (c) 441 (d) 401
3. 3.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2411 (b) 2499 (c) 2401 (d) 2601
4. 4.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 11)
5. 5.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 899 (c) 851 (d) 961
6. 6.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 4 (c) 2 (d) 130
7. 7.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 3025 (c) 2025 (d) 2475
8. 8.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 478 (c) 324 (d) 170
9. 9.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 3025 (c) 3035 (d) 4225
10. 10.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
11. 11.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1681 (c) 1599 (d) 1531
12. 12.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 11) (c) (x + 1)(x + 32) (d) (x − 4)(x − 8)
13. 13.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 11 (c) 12 (d) 18
14. 14.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
15. 15.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 5 (c) 361 (d) 25
16. 16.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
17. 17.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 4 (d) 2
18. 18.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 52 (c) 100 (d) 76
19. 19.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x − 2)(x − 7) (c) (x + 1)(x + 14) (d) (x + 1)(x + 8)
20. 20.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 14 (c) 13 (d) 42
21. 21.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 4) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
22. 22.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2116 (c) 2916 (d) 2126
23. 23.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)
24. 24.What are the roots of x² − 10x + 24 = 0? (a) 4 and 6 (b) 1 and 24 (c) 2 and 12 (d) 1 and 9
25. 25.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 625 (c) 225 (d) 375
26. 26.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 8 (c) 10 (d) 9
27. 27.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)
28. 28.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
29. 29.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 686 (c) 1156 (d) 676
30. 30.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 14 (c) 1 and 54 (d) 6 and 9
31. 31.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
32. 32.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 63 (d) 62
33. 33.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 15 (c) 48 (d) 14
34. 34.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 78 (c) 57 (d) 50
35. 35.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
36. 36.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 1 (c) 113 (d) 169
37. 37.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 185 (d) 273
38. 38.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 4 (c) 74 (d) 144
39. 39.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 333 (d) 225
40. 40.What are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 4 and 9 (c) 1 and 36 (d) 1 and 12

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