Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 684 · 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.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 169 (c) 85 (d) 127
2. 2.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 8096 (c) 8464 (d) 7754
3. 3.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 36 (c) 46 (d) 26
4. 4.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 57 (d) 56
5. 5.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 15 (c) 13 (d) 14
6. 6.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 169 (c) 129 (d) 89
7. 7.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 289 (c) 5 (d) 25
8. 8.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 1)(x + 12) (d) (x + 5)(x + 8)
9. 9.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4225 (b) 3625 (c) 3025 (d) 4235
10. 10.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 10025 (c) 11025 (d) 11035
11. 11.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 576 (d) 416
12. 12.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
13. 13.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 24) (c) (x + 1)(x + 10) (d) (x + 3)(x + 8)
14. 14.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 400 (c) 16 (d) 208
15. 15.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 9 (c) 225 (d) 3
16. 16.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 193 (c) 382 (d) 130
17. 17.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
18. 18.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 2)(x + 6) (c) (x − 2)(x − 6) (d) (x + 1)(x + 12)
19. 19.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
20. 20.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 2)(x + 9) (c) (x − 2)(x − 9) (d) (x + 1)(x + 18)
21. 21.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 1 and 16 (d) 8 and 9
22. 22.If a + b = 13 and ab = 40, what is (a − b)²? (a) 9 (b) 3 (c) 169 (d) 89
23. 23.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 4904 (c) 5184 (d) 5194
24. 24.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 79 (c) 142 (d) 100
25. 25.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 1 and 16 (c) 2 and 8 (d) 1 and 9
26. 26.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1034 (c) 1024 (d) 904
27. 27.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 5 (c) 4 (d) 6
28. 28.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 5184 (c) 4634 (d) 4624
29. 29.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
30. 30.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4366 (c) 4884 (d) 4356
31. 31.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
32. 32.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 17 (c) 19 (d) 11
33. 33.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 433 (d) 145
34. 34.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 10404 (c) 9996 (d) 9604
35. 35.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 225 (d) 441
36. 36.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 11 (c) 16 (d) 9
37. 37.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
38. 38.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 289 (c) 299 (d) 391
39. 39.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
40. 40.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x + 1)(x + 24)

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad