Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 106 · 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 18²? (a) 484 (b) 324 (c) 334 (d) 396
2. 2.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)
3. 3.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1296 (c) 1946 (d) 1936
4. 4.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 2 (c) 100 (d) 52
5. 5.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1521 (c) 1531 (d) 1681
6. 6.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)
7. 7.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 85 (c) 1 (d) 127
8. 8.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1691 (d) 1681
9. 9.What are the roots of x² − 7x + 12 = 0? (a) 2 and 6 (b) 3 and 4 (c) 1 and 6 (d) 1 and 12
10. 10.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
11. 11.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10816 (c) 10016 (d) 10826
12. 12.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1454 (c) 1764 (d) 1596
13. 13.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6571 (c) 6561 (d) 6241
14. 14.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
15. 15.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 625 (c) 425 (d) 225
16. 16.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 26 (c) 31 (d) 46
17. 17.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
18. 18.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
19. 19.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
20. 20.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 33 (c) 17 (d) 21
21. 21.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
22. 22.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4489 (c) 5329 (d) 4909
23. 23.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x + 1)(x + 40) (d) (x − 5)(x − 8)
24. 24.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 916 (c) 1166 (d) 1156
25. 25.What is the product of the roots of x² − 11x + 28 = 0? (a) 28 (b) 11 (c) 27 (d) 29
26. 26.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 93 (c) 177 (d) 121
27. 27.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 121 (c) 25 (d) 5
28. 28.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 15 (c) 1 and 7 (d) 3 and 5
29. 29.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 1)(x + 24) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
30. 30.If a + b = 13 and ab = 40, what is (a − b)²? (a) 9 (b) 169 (c) 3 (d) 89
31. 31.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 4 (c) 256 (d) 136
32. 32.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 196 (c) 276 (d) 156
33. 33.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 42 (c) 13 (d) 43
34. 34.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 10201 (c) 9811 (d) 9801
35. 35.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 2 and 8 (c) 1 and 9 (d) 3 and 7
36. 36.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 14)
37. 37.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 46 (c) 45 (d) 44
38. 38.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 16 (c) 4 (d) 106
39. 39.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 324 (d) 484
40. 40.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 2025 (c) 1625 (d) 1225

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