Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 912 · 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.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 1 and 36 (c) 2 and 18 (d) 4 and 9
2. 2.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
3. 3.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 6 (c) 146 (d) 36
4. 4.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)
5. 5.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 54 (d) 53
6. 6.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 36 (c) 28 (d) 20
7. 7.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 19 (d) 21
8. 8.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 25 (c) 9 (d) 3
9. 9.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7754 (c) 7744 (d) 8096
10. 10.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 10 (c) 9 (d) 11
11. 11.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
12. 12.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
13. 13.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2035 (c) 1225 (d) 2025
14. 14.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
15. 15.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 57 (c) 55 (d) 15
16. 16.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 1 and 16 (d) 2 and 36
17. 17.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 217 (c) 145 (d) 1
18. 18.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 2)(x + 7) (c) (x − 2)(x − 7) (d) (x + 1)(x + 8)
19. 19.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7066 (c) 7056 (d) 5776
20. 20.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 22 (c) 13 (d) 10
21. 21.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5486 (c) 5476 (d) 4916
22. 22.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 74 (c) 4 (d) 2
23. 23.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)
24. 24.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 9 (c) 3 (d) 185
25. 25.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3844 (c) 3374 (d) 3596
26. 26.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6404 (c) 6724 (d) 6084
27. 27.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 148 (c) 196 (d) 100
28. 28.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 8 (c) 7 (d) 5
29. 29.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6561 (c) 6399 (d) 6241
30. 30.What is the product of the roots of x² − 12x + 27 = 0? (a) 12 (b) 26 (c) 27 (d) 28
31. 31.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 4 and 7 (c) 2 and 14 (d) 1 and 10
32. 32.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1591 (c) 1849 (d) 1379
33. 33.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4884 (c) 5476 (d) 4356
34. 34.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 5194 (c) 5184 (d) 4904
35. 35.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
36. 36.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 4 (c) 170 (d) 324
37. 37.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1681 (c) 1691 (d) 1601
38. 38.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
39. 39.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2484 (c) 2126 (d) 2116
40. 40.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 61 (c) 41 (d) 81

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