Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 178 · 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 29²? (a) 961 (b) 899 (c) 851 (d) 841
2. 2.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 49 (c) 73 (d) 37
3. 3.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 10 (c) 12 (d) 11
4. 4.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 686 (c) 676 (d) 884
5. 5.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
6. 6.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 334 (c) 484 (d) 396
7. 7.If a + b = 16 and ab = 55, what is (a − b)²? (a) 36 (b) 6 (c) 256 (d) 146
8. 8.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
9. 9.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3259 (c) 3591 (d) 3249
10. 10.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 148 (c) 196 (d) 100
11. 11.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
12. 12.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 3 and 4 (c) 2 and 6 (d) 1 and 12
13. 13.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 8125 (c) 9025 (d) 7225
14. 14.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
15. 15.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6084 (c) 6724 (d) 6734
16. 16.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 55 (d) 15
17. 17.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 217 (c) 289 (d) 145
18. 18.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1444 (c) 1764 (d) 1596
19. 19.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 9 (c) 15 (d) 7
20. 20.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 7931 (c) 8281 (d) 8099
21. 21.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 4 (c) 144 (d) 16
22. 22.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
23. 23.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 484 (d) 494
24. 24.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
25. 25.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6561 (c) 6241 (d) 6399
26. 26.What is the product of the roots of x² − 6x + 8 = 0? (a) 8 (b) 6 (c) 9 (d) 7
27. 27.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
28. 28.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 416 (d) 576
29. 29.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 1 (c) 91 (d) 61
30. 30.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
31. 31.What is the product of the roots of x² − 10x + 21 = 0? (a) 21 (b) 20 (c) 10 (d) 22
32. 32.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10009 (c) 10619 (d) 10609
33. 33.What are the roots of x² − 10x + 24 = 0? (a) 4 and 6 (b) 1 and 9 (c) 1 and 24 (d) 2 and 12
34. 34.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 62 (d) 64
35. 35.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 8 (c) 4 and 5 (d) 1 and 20
36. 36.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 7396 (c) 8084 (d) 8836
37. 37.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 43 (c) 61 (d) 37
38. 38.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 2 and 36 (d) 8 and 9
39. 39.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 196 (c) 156 (d) 116
40. 40.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)

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