Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 736 · 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 = 18 and ab = 77, what is a² + b²? (a) 324 (b) 478 (c) 170 (d) 247
2. 2.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
3. 3.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
4. 4.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 20 (c) 9 (d) 19
5. 5.If a + b = 16 and ab = 55, what is (a − b)²? (a) 36 (b) 256 (c) 146 (d) 6
6. 6.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 208 (c) 592 (d) 400
7. 7.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 10 (c) 16 (d) 4
8. 8.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 1 (c) 61 (d) 121
9. 9.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 5 and 9 (c) 3 and 15 (d) 1 and 13
10. 10.If a + b = 16 and ab = 48, what is a² + b²? (a) 160 (b) 352 (c) 256 (d) 208
11. 11.If a + b = 11 and ab = 28, what is a² + b²? (a) 177 (b) 65 (c) 93 (d) 121
12. 12.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 22 (c) 10 (d) 13
13. 13.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 36 (c) 16 (d) 4
14. 14.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 148 (c) 196 (d) 244
15. 15.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4875 (c) 4225 (d) 4235
16. 16.What are the roots of x² − 8x + 12 = 0? (a) 2 and 6 (b) 3 and 4 (c) 1 and 7 (d) 1 and 12
17. 17.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 81 (c) 225 (d) 9
18. 18.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 9 (c) 1 (d) 5
19. 19.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 1 and 15 (c) 2 and 6 (d) 3 and 5
20. 20.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2501 (c) 2611 (d) 2601
21. 21.If a + b = 13 and ab = 22, what is a² + b²? (a) 125 (b) 213 (c) 147 (d) 169
22. 22.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 100 (c) 82 (d) 91
23. 23.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 117 (c) 225 (d) 3
24. 24.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 145 (d) 289
25. 25.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 11 (c) 12 (d) 35
26. 26.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
27. 27.If a + b = 9 and ab = 8, what is (a − b)²? (a) 65 (b) 81 (c) 49 (d) 7
28. 28.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 30 (c) 1 and 10 (d) 2 and 15
29. 29.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 484 (c) 334 (d) 324
30. 30.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 3 (c) 29 (d) 9
31. 31.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 961 (c) 841 (d) 851
32. 32.If a + b = 9 and ab = 14, what is (a − b)²? (a) 25 (b) 81 (c) 53 (d) 5
33. 33.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x + 1)(x + 16) (d) (x − 2)(x − 8)
34. 34.What are the roots of x² − 11x + 24 = 0? (a) 3 and 8 (b) 2 and 12 (c) 1 and 10 (d) 1 and 24
35. 35.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6724 (c) 6084 (d) 6396
36. 36.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 193 (c) 529 (d) 361
37. 37.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 1225 (c) 2035 (d) 2025
38. 38.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 6 (c) 196 (d) 36
39. 39.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
40. 40.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 8075 (b) 7235 (c) 9025 (d) 7225

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