Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 749 · 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.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
2. 2.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
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 = 15 and ab = 54, what is a² + b²? (a) 171 (b) 117 (c) 225 (d) 333
5. 5.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3035 (d) 3025
6. 6.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10211 (c) 10201 (d) 10001
7. 7.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
8. 8.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2401 (c) 2411 (d) 2499
9. 9.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 484 (c) 324 (d) 164
10. 10.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6399 (c) 6561 (d) 6251
11. 11.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 2 and 36 (d) 1 and 72
12. 12.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 13 (c) 14 (d) 15
13. 13.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 64 (c) 104 (d) 8
14. 14.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
15. 15.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7396 (c) 7406 (d) 8836
16. 16.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)
17. 17.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 117 (c) 9 (d) 225
18. 18.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
19. 19.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 10 (c) 7 (d) 6
20. 20.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3491 (b) 3721 (c) 3599 (d) 3481
21. 21.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 4225 (c) 3025 (d) 3035
22. 22.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 193 (c) 145 (d) 157
23. 23.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
24. 24.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 325 (c) 125 (d) 225
25. 25.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1584 (c) 1936 (d) 1296
26. 26.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
27. 27.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 225 (c) 333 (d) 657
28. 28.If a + b = 10 and ab = 16, what is (a − b)²? (a) 36 (b) 68 (c) 100 (d) 6
29. 29.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5184 (c) 5194 (d) 4624
30. 30.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 289 (c) 7 (d) 49
31. 31.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
32. 32.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1764 (c) 1454 (d) 1596
33. 33.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3609 (c) 3249 (d) 3969
34. 34.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 11 (c) 12 (d) 35
35. 35.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
36. 36.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 2 (d) 4
37. 37.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 1 and 18 (c) 1 and 10 (d) 2 and 9
38. 38.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 17 (d) 72
39. 39.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1601 (c) 1691 (d) 1681
40. 40.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 8 (d) 6

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