Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 979 · 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 76²? (a) 5786 (b) 5776 (c) 6384 (d) 7056
2. 2.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 6 (c) 5 (d) 8
3. 3.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
4. 4.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 14 (c) 15 (d) 16
5. 5.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 34 (c) 2 (d) 4
6. 6.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 1)(x + 28)
7. 7.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
8. 8.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
9. 9.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)
10. 10.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 2 and 5 (c) 1 and 5 (d) 2 and 4
11. 11.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 5476 (c) 4366 (d) 4884
12. 12.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
13. 13.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 14 (c) 16 (d) 56
14. 14.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
15. 15.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6889 (c) 5929 (d) 6409
16. 16.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
17. 17.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 2 and 16 (c) 1 and 32 (d) 4 and 8
18. 18.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 91 (c) 1 (d) 121
19. 19.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10826 (c) 10816 (d) 10016
20. 20.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
21. 21.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 289 (c) 3 (d) 9
22. 22.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1296 (c) 1616 (d) 1936
23. 23.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 193 (c) 25 (d) 361
24. 24.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 193 (c) 361 (d) 529
25. 25.If a + b = 8 and ab = 7, what is (a − b)²? (a) 36 (b) 64 (c) 50 (d) 6
26. 26.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 63 (c) 117 (d) 45
27. 27.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 11 (c) 21 (d) 9
28. 28.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 64 (c) 40 (d) 52
29. 29.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1601 (c) 1691 (d) 1681
30. 30.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 56 (d) 1 and 14
31. 31.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 3 and 4 (c) 1 and 12 (d) 2 and 6
32. 32.What is the product of the roots of x² − 12x + 32 = 0? (a) 32 (b) 31 (c) 12 (d) 33
33. 33.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 8)
34. 34.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 8464 (c) 8096 (d) 7754
35. 35.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 4761 (c) 5041 (d) 5051
36. 36.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3721 (c) 3601 (d) 3731
37. 37.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 37 (c) 35 (d) 36
38. 38.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 256 (c) 586 (d) 416
39. 39.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 625 (c) 225 (d) 235
40. 40.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2819 (c) 2809 (d) 2209

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