Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 414 · 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 = 11 and ab = 18, what is a² + b²? (a) 103 (b) 157 (c) 121 (d) 85
2. 2.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 7 and 9 (d) 3 and 21
3. 3.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 49 (c) 34 (d) 94
4. 4.What are the roots of x² − 11x + 28 = 0? (a) 1 and 10 (b) 2 and 14 (c) 4 and 7 (d) 1 and 28
5. 5.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)
6. 6.If a + b = 6 and ab = 5, what is (a − b)²? (a) 4 (b) 26 (c) 16 (d) 36
7. 7.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 43 (c) 13 (d) 41
8. 8.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7225 (b) 7235 (c) 5625 (d) 6425
9. 9.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 729 (c) 1089 (d) 1099
10. 10.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 12 (c) 28 (d) 26
11. 11.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 409 (c) 529 (d) 539
12. 12.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
13. 13.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 441 (c) 371 (d) 399
14. 14.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 26 (c) 31 (d) 46
15. 15.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
16. 16.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1599 (c) 1681 (d) 1531
17. 17.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
18. 18.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 169 (c) 97 (d) 241
19. 19.If a + b = 11 and ab = 10, what is a² + b²? (a) 101 (b) 121 (c) 141 (d) 111
20. 20.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5051 (c) 4901 (d) 5041
21. 21.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
22. 22.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 1 and 12 (c) 1 and 40 (d) 5 and 8
23. 23.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 1 and 14 (d) 7 and 8
24. 24.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 80 (c) 112 (d) 144
25. 25.What is the sum of the roots of x² − 11x + 28 = 0? (a) 11 (b) 10 (c) 28 (d) 12
26. 26.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 45 (c) 81 (d) 63
27. 27.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
28. 28.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 9 (c) 18 (d) 8
29. 29.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
30. 30.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 34 (c) 2 (d) 4
31. 31.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 2 (c) 36 (d) 4
32. 32.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
33. 33.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 10 (c) 9 (d) 8
34. 34.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
35. 35.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 4)(x + 5) (c) (x + 1)(x + 20) (d) (x − 4)(x − 5)
36. 36.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 1625 (c) 2035 (d) 2025
37. 37.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 6 and 8 (c) 2 and 24 (d) 1 and 13
38. 38.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5329 (b) 5339 (c) 4489 (d) 4909
39. 39.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
40. 40.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 61 (c) 91 (d) 121

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