Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 287 · 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.If a + b = 13 and ab = 40, what is (a − b)²? (a) 9 (b) 89 (c) 169 (d) 3
2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
3. 3.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 6391 (c) 6889 (d) 5929
4. 4.What is the product of the roots of x² − 8x + 12 = 0? (a) 11 (b) 8 (c) 12 (d) 13
5. 5.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 55 (d) 57
6. 6.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 971 (c) 961 (d) 901
7. 7.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x + 1)(x + 18) (d) (x − 3)(x − 6)
8. 8.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6409 (c) 6889 (d) 5929
9. 9.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 1225 (c) 875 (d) 635
10. 10.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 1 and 36 (c) 4 and 9 (d) 2 and 18
11. 11.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 235 (c) 225 (d) 625
12. 12.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 6 and 8 (c) 1 and 48 (d) 1 and 13
13. 13.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 36 (c) 20 (d) 52
14. 14.If a + b = 16 and ab = 55, what is (a − b)²? (a) 36 (b) 6 (c) 146 (d) 256
15. 15.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
16. 16.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
17. 17.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 64 (c) 8 (d) 104
18. 18.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
19. 19.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 7235 (c) 7225 (d) 8075
20. 20.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x + 2)(x + 3)
21. 21.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 63 (c) 16 (d) 64
22. 22.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 334 (c) 324 (d) 396
23. 23.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 14 (c) 15 (d) 16
24. 24.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 12)
25. 25.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
26. 26.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 127 (c) 169 (d) 1
27. 27.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 10 (c) 22 (d) 21
28. 28.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 400 (d) 4
29. 29.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 8 and 9 (d) 1 and 16
30. 30.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 1 (d) 289
31. 31.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2304 (c) 2504 (d) 2714
32. 32.What is the sum of the roots of x² − 14x + 45 = 0? (a) 14 (b) 45 (c) 13 (d) 15
33. 33.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 361 (c) 181 (d) 271
34. 34.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x − 3)(x − 7) (c) (x + 1)(x + 9) (d) (x + 3)(x + 7)
35. 35.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 2035 (c) 1625 (d) 1225
36. 36.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 223 (c) 289 (d) 157
37. 37.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 5 and 8 (c) 1 and 12 (d) 2 and 20
38. 38.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 271 (d) 541
39. 39.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 382 (c) 256 (d) 130
40. 40.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 7 and 9 (c) 1 and 15 (d) 3 and 21

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