Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 145 · 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 35²? (a) 1575 (b) 1225 (c) 1235 (d) 2025
2. 2.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
3. 3.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 256 (c) 16 (d) 4
4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
5. 5.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 10004 (c) 10404 (d) 9604
6. 6.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 13 (c) 1 and 48 (d) 6 and 8
7. 7.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 17 (d) 72
8. 8.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
9. 9.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 484 (c) 334 (d) 324
10. 10.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 337 (c) 169 (d) 113
11. 11.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 193 (c) 130 (d) 256
12. 12.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 8 (c) 160 (d) 64
13. 13.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 56 (c) 1 and 14 (d) 2 and 28
14. 14.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 851 (c) 841 (d) 899
15. 15.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x + 1)(x + 15)
16. 16.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 8125 (c) 9035 (d) 7225
17. 17.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 45 (c) 13 (d) 14
18. 18.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 249 (c) 129 (d) 89
19. 19.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)
20. 20.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x − 4)(x − 8)
21. 21.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 5184 (c) 4624 (d) 4634
22. 22.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 1235 (c) 925 (d) 1225
23. 23.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 30 (c) 29 (d) 11
24. 24.What is the sum of the roots of x² − 9x + 18 = 0? (a) 18 (b) 9 (c) 8 (d) 10
25. 25.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 5625 (c) 4225 (d) 4925
26. 26.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
27. 27.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 400 (c) 304 (d) 208
28. 28.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 16 (c) 10 (d) 11
29. 29.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 2025 (c) 3025 (d) 2475
30. 30.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
31. 31.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 181 (c) 61 (d) 121
32. 32.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 8 (c) 7 (d) 5
33. 33.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 784 (c) 904 (d) 1034
34. 34.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
35. 35.If a + b = 7 and ab = 10, what is (a − b)²? (a) 3 (b) 49 (c) 9 (d) 29
36. 36.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 324 (c) 6 (d) 36
37. 37.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
38. 38.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 1 and 24 (c) 3 and 8 (d) 1 and 10
39. 39.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)
40. 40.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 8 (c) 15 (d) 9

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