Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 148 · 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 31²? (a) 901 (b) 961 (c) 841 (d) 971
2. 2.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)
3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
4. 4.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 136 (c) 376 (d) 196
5. 5.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)
6. 6.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 10 (c) 24 (d) 9
7. 7.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4891 (b) 4489 (c) 4499 (d) 5329
8. 8.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 271 (d) 181
9. 9.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6409 (c) 6899 (d) 6889
10. 10.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)
11. 11.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 249 (c) 89 (d) 169
12. 12.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 400 (c) 304 (d) 208
13. 13.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3025 (c) 3035 (d) 2025
14. 14.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
15. 15.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 172 (c) 148 (d) 196
16. 16.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 225 (d) 9
17. 17.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 2035 (c) 3025 (d) 2025
18. 18.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3854 (c) 3844 (d) 3364
19. 19.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x − 4)(x − 6)
20. 20.If a + b = 12 and ab = 27, what is (a − b)²? (a) 144 (b) 90 (c) 6 (d) 36
21. 21.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10404 (c) 9604 (d) 10414
22. 22.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1444 (c) 1596 (d) 1454
23. 23.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 313 (c) 137 (d) 181
24. 24.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)
25. 25.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9419 (c) 9991 (d) 9409
26. 26.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 164 (c) 324 (d) 2
27. 27.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 3 (c) 121 (d) 9
28. 28.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 12 (c) 42 (d) 14
29. 29.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1936 (c) 1616 (d) 1296
30. 30.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 391 (c) 299 (d) 289
31. 31.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 1 and 10 (c) 1 and 28 (d) 2 and 14
32. 32.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 2)(x + 5) (c) (x + 1)(x + 10) (d) (x + 1)(x + 6)
33. 33.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 34 (c) 36 (d) 35
34. 34.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 23 (c) 10 (d) 25
35. 35.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9999 (c) 9811 (d) 9801
36. 36.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 41 (c) 81 (d) 61
37. 37.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)
38. 38.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3374 (c) 3596 (d) 3844
39. 39.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1681 (c) 1521 (d) 1599
40. 40.If a + b = 11 and ab = 18, what is a² + b²? (a) 103 (b) 157 (c) 121 (d) 85

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