Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 187 · 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 49²? (a) 2401 (b) 2499 (c) 2601 (d) 2411
2. 2.If a + b = 11 and ab = 24, what is a² + b²? (a) 73 (b) 97 (c) 169 (d) 121
3. 3.What is the product of the roots of x² − 13x + 40 = 0? (a) 39 (b) 13 (c) 41 (d) 40
4. 4.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 10025 (c) 11035 (d) 9025
5. 5.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 382 (c) 130 (d) 256
6. 6.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 10001 (c) 9801 (d) 10211
7. 7.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 1 and 16 (d) 8 and 9
8. 8.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x + 2)(x + 8) (d) (x − 2)(x − 8)
9. 9.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
10. 10.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 12 (c) 11 (d) 30
11. 11.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
12. 12.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 301 (c) 202 (d) 400
13. 13.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 361 (c) 25 (d) 193
14. 14.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)
15. 15.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 494 (c) 404 (d) 484
16. 16.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 9 (c) 185 (d) 3
17. 17.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 24 (c) 23 (d) 11
18. 18.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 196 (c) 16 (d) 106
19. 19.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
20. 20.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 181 (c) 137 (d) 313
21. 21.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 109 (c) 214 (d) 144
22. 22.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 576 (d) 416
23. 23.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3481 (c) 3731 (d) 3601
24. 24.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1369 (c) 1609 (d) 1849
25. 25.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 11 (c) 30 (d) 29
26. 26.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 3616 (c) 4096 (d) 3136
27. 27.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 4901 (c) 5051 (d) 5041
28. 28.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 10) (c) (x + 2)(x + 9) (d) (x − 2)(x − 9)
29. 29.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
30. 30.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 2 and 4 (c) 1 and 4 (d) 2 and 3
31. 31.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
32. 32.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 48 (c) 13 (d) 14
33. 33.What are the roots of x² − 9x + 14 = 0? (a) 1 and 14 (b) 1 and 8 (c) 3 and 6 (d) 2 and 7
34. 34.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 82 (c) 91 (d) 100
35. 35.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 11)
36. 36.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 33 (c) 31 (d) 32
37. 37.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 441 (c) 657 (d) 225
38. 38.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 11 (c) 10 (d) 24
39. 39.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 180 (c) 252 (d) 324
40. 40.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)

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