Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 65 · 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 75²? (a) 5625 (b) 4225 (c) 5635 (d) 4925
2. 2.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 13 (c) 36 (d) 14
3. 3.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 193 (c) 130 (d) 382
4. 4.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 3136 (c) 4106 (d) 4096
5. 5.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)
6. 6.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 4 (c) 16 (d) 400
7. 7.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 214 (c) 109 (d) 74
8. 8.What is the sum of the roots of x² − 11x + 28 = 0? (a) 11 (b) 12 (c) 28 (d) 10
9. 9.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 18 (c) 17 (d) 9
10. 10.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
11. 11.If a + b = 12 and ab = 11, what is a² + b²? (a) 122 (b) 166 (c) 133 (d) 144
12. 12.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
13. 13.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6401 (c) 6561 (d) 6241
14. 14.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 25 (c) 5 (d) 125
15. 15.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1296 (c) 1936 (d) 1946
16. 16.If a + b = 13 and ab = 30, what is (a − b)²? (a) 49 (b) 7 (c) 169 (d) 109
17. 17.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1306 (c) 1936 (d) 1584
18. 18.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
19. 19.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 64 (c) 34 (d) 4
20. 20.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 625 (c) 375 (d) 225
21. 21.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 2 and 14 (c) 1 and 28 (d) 1 and 10
22. 22.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 181 (d) 361
23. 23.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 8649 (c) 7569 (d) 8659
24. 24.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 1 and 36 (c) 4 and 9 (d) 2 and 18
25. 25.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)
26. 26.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4634 (c) 4896 (d) 4624
27. 27.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)
28. 28.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2025 (c) 1625 (d) 2035
29. 29.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
30. 30.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 16 (d) 18
31. 31.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 7 (c) 1 (d) 5
32. 32.If a + b = 11 and ab = 10, what is (a − b)²? (a) 121 (b) 101 (c) 81 (d) 9
33. 33.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 5625 (c) 4235 (d) 4225
34. 34.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 14 (c) 16 (d) 56
35. 35.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
36. 36.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6084 (c) 6094 (d) 6724
37. 37.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 196 (c) 163 (d) 130
38. 38.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
39. 39.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 11 (c) 12 (d) 18
40. 40.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 81 (c) 41 (d) 1

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