Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 815 · 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 39²? (a) 1681 (b) 1521 (c) 1599 (d) 1531
2. 2.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1604 (c) 1764 (d) 1444
3. 3.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 100 (c) 82 (d) 91
4. 4.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 11 (c) 9 (d) 10
5. 5.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 52 (c) 100 (d) 2
6. 6.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 42 (c) 13 (d) 43
7. 7.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
8. 8.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 7 (c) 169 (d) 289
9. 9.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 181 (c) 121 (d) 91
10. 10.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3601 (c) 3721 (d) 3481
11. 11.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 39 (c) 40 (d) 41
12. 12.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 5 (c) 361 (d) 193
13. 13.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 289 (c) 391 (d) 529
14. 14.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 125 (c) 225 (d) 25
15. 15.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
16. 16.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
17. 17.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 4 (c) 400 (d) 208
18. 18.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 13 (d) 14
19. 19.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
20. 20.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 256 (c) 16 (d) 4
21. 21.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 130 (c) 382 (d) 256
22. 22.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 5 (c) 6 (d) 8
23. 23.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 598 (c) 301 (d) 202
24. 24.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
25. 25.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 451 (c) 441 (d) 361
26. 26.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 399 (c) 361 (d) 371
27. 27.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 1 and 8 (c) 2 and 9 (d) 3 and 6
28. 28.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2714 (c) 2704 (d) 2504
29. 29.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 9) (c) (x + 1)(x + 21) (d) (x − 3)(x − 7)
30. 30.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 18 (c) 9 (d) 8
31. 31.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
32. 32.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 21 (c) 9 (d) 10
33. 33.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 225 (c) 3 (d) 9
34. 34.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 3025 (c) 4225 (d) 3625
35. 35.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
36. 36.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 3025 (c) 2035 (d) 2475
37. 37.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 32 (c) 31 (d) 33
38. 38.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 529 (c) 409 (d) 539
39. 39.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 7225 (c) 9025 (d) 8075
40. 40.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)

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