Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 633 · 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 91²? (a) 8291 (b) 8101 (c) 7921 (d) 8281
2. 2.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 21 (c) 10 (d) 9
3. 3.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 1 (d) 145
4. 4.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2926 (c) 2916 (d) 2116
5. 5.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
6. 6.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 289 (c) 409 (d) 169
7. 7.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 625 (c) 875 (d) 1225
8. 8.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 5184 (c) 4624 (d) 4904
9. 9.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3575 (c) 3025 (d) 3035
10. 10.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 9801 (c) 10201 (d) 10001
11. 11.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 9419 (c) 10609 (d) 9991
12. 12.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 6 and 8 (c) 1 and 48 (d) 1 and 13
13. 13.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 7 (c) 6 (d) 8
14. 14.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 49 (c) 7 (d) 169
15. 15.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 11 (c) 13 (d) 12
16. 16.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 57 (d) 15
17. 17.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 4 (d) 16
18. 18.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
19. 19.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 100 (c) 196 (d) 2
20. 20.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8109 (c) 8659 (d) 8649
21. 21.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
22. 22.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 21 (c) 22 (d) 10
23. 23.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 54 (c) 2 and 27 (d) 1 and 14
24. 24.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 1 (c) 19 (d) 13
25. 25.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 9 (c) 153 (d) 81
26. 26.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 40 (c) 41 (d) 39
27. 27.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x − 2)(x − 4) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
28. 28.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 15)
29. 29.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x + 2)(x + 5) (d) (x − 2)(x − 5)
30. 30.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 90 (c) 198 (d) 144
31. 31.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 9216 (c) 10016 (d) 10816
32. 32.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 225 (c) 375 (d) 235
33. 33.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
34. 34.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1521 (c) 1681 (d) 1601
35. 35.What is the product of the roots of x² − 12x + 35 = 0? (a) 35 (b) 36 (c) 34 (d) 12
36. 36.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 576 (c) 256 (d) 384
37. 37.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)
38. 38.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 10 (c) 18 (d) 12
39. 39.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6724 (c) 6734 (d) 6404
40. 40.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 14 (d) 13

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