Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 491 · 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 = 16 and ab = 55, what is a² + b²? (a) 256 (b) 201 (c) 366 (d) 146
2. 2.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 100 (c) 2 (d) 196
3. 3.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 34 (c) 64 (d) 4
4. 4.What is the product of the roots of x² − 9x + 18 = 0? (a) 17 (b) 19 (c) 18 (d) 9
5. 5.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1616 (c) 1296 (d) 1936
6. 6.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 13 (c) 37 (d) 19
7. 7.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 4899 (c) 4761 (d) 5041
8. 8.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 13 (c) 7 (d) 12
9. 9.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3596 (c) 3374 (d) 3364
10. 10.If a + b = 9 and ab = 14, what is (a − b)²? (a) 81 (b) 5 (c) 25 (d) 53
11. 11.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 36 (c) 6 (d) 144
12. 12.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1774 (c) 1764 (d) 1444
13. 13.If a + b = 7 and ab = 6, what is a² + b²? (a) 61 (b) 37 (c) 49 (d) 43
14. 14.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 127 (c) 85 (d) 253
15. 15.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 592 (c) 400 (d) 304
16. 16.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 7921 (c) 8291 (d) 8101
17. 17.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
18. 18.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1521 (c) 1681 (d) 1601
19. 19.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2025 (c) 2035 (d) 1625
20. 20.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 2 and 21 (c) 1 and 42 (d) 1 and 12
21. 21.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 896 (c) 794 (d) 784
22. 22.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 27 (c) 13 (d) 12
23. 23.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 429 (c) 219 (d) 149
24. 24.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 382 (c) 193 (d) 130
25. 25.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 15 (c) 1 and 63 (d) 7 and 9
26. 26.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 256 (c) 130 (d) 2
27. 27.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 16 (c) 56 (d) 15
28. 28.If a + b = 8 and ab = 7, what is a² + b²? (a) 78 (b) 50 (c) 64 (d) 57
29. 29.What is the product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 9 (c) 8 (d) 6
30. 30.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 271 (d) 181
31. 31.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1584 (c) 1296 (d) 1936
32. 32.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 72 (d) 71
33. 33.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 10 (c) 18 (d) 11
34. 34.If a + b = 9 and ab = 8, what is (a − b)²? (a) 81 (b) 7 (c) 49 (d) 65
35. 35.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x + 1)(x + 7) (d) (x − 3)(x − 5)
36. 36.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9996 (c) 9604 (d) 9614
37. 37.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 12 (c) 32 (d) 13
38. 38.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
39. 39.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 157 (c) 193 (d) 145
40. 40.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 4 (c) 208 (d) 400

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