Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 212 · 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 = 12 and ab = 20, what is a² + b²? (a) 124 (b) 144 (c) 104 (d) 184
2. 2.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
3. 3.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 484 (d) 494
4. 4.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)
5. 5.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
6. 6.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1034 (c) 784 (d) 1024
7. 7.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 13 (c) 15 (d) 14
8. 8.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 71 (d) 17
9. 9.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5184 (c) 5194 (d) 4624
10. 10.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 214 (c) 74 (d) 144
11. 11.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 1 and 56 (d) 7 and 8
12. 12.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 6 (c) 36 (d) 146
13. 13.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 1 and 16 (d) 8 and 9
14. 14.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 3025 (c) 2025 (d) 2525
15. 15.If a + b = 13 and ab = 12, what is a² + b²? (a) 193 (b) 157 (c) 169 (d) 145
16. 16.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x − 3)(x − 4)
17. 17.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 333 (c) 225 (d) 441
18. 18.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2916 (c) 2126 (d) 2116
19. 19.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)
20. 20.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 30 (c) 31 (d) 11
21. 21.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 121 (c) 41 (d) 81
22. 22.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 26 (c) 4 (d) 16
23. 23.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 15 (d) 63
24. 24.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3374 (c) 3844 (d) 3364
25. 25.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 100 (c) 58 (d) 79
26. 26.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 18 (d) 72
27. 27.If a + b = 8 and ab = 15, what is (a − b)²? (a) 4 (b) 2 (c) 64 (d) 34
28. 28.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 144 (c) 208 (d) 80
29. 29.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 17 (c) 11 (d) 19
30. 30.If a + b = 14 and ab = 24, what is (a − b)²? (a) 100 (b) 10 (c) 148 (d) 196
31. 31.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 1225 (c) 2035 (d) 2025
32. 32.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9999 (c) 9811 (d) 10201
33. 33.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 12 (c) 42 (d) 14
34. 34.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 24 (c) 11 (d) 10
35. 35.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
36. 36.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 13 (c) 48 (d) 15
37. 37.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 400 (c) 2 (d) 4
38. 38.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
39. 39.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 4 (c) 256 (d) 2
40. 40.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 30 (c) 11 (d) 12

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