Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 863 · 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 + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
2. 2.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 361 (c) 371 (d) 399
3. 3.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 136 (c) 16 (d) 4
4. 4.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
5. 5.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1601 (d) 1681
6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
7. 7.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4489 (c) 4499 (d) 4891
8. 8.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 4 (c) 208 (d) 16
9. 9.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 8 (d) 7
10. 10.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1604 (c) 1774 (d) 1764
11. 11.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
12. 12.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
13. 13.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
14. 14.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
15. 15.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
16. 16.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 11 (c) 29 (d) 28
17. 17.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 1 and 32 (c) 2 and 16 (d) 4 and 8
18. 18.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 8464 (c) 8096 (d) 7754
19. 19.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)
20. 20.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 72 (d) 17
21. 21.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 49 (c) 289 (d) 169
22. 22.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 3 and 21 (c) 1 and 15 (d) 1 and 63
23. 23.If a + b = 8 and ab = 7, what is a² + b²? (a) 50 (b) 78 (c) 64 (d) 57
24. 24.If a + b = 11 and ab = 10, what is (a − b)²? (a) 121 (b) 101 (c) 81 (d) 9
25. 25.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 2 and 36 (d) 1 and 72
26. 26.What are the roots of x² − 10x + 24 = 0? (a) 4 and 6 (b) 2 and 12 (c) 1 and 24 (d) 1 and 9
27. 27.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 14 (c) 56 (d) 16
28. 28.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 79 (c) 100 (d) 58
29. 29.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 116 (c) 6 (d) 36
30. 30.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 256 (c) 6 (d) 36
31. 31.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2491 (c) 2209 (d) 2219
32. 32.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 9996 (c) 10404 (d) 9614
33. 33.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 144 (c) 80 (d) 112
34. 34.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)
35. 35.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 1 and 14 (c) 2 and 28 (d) 7 and 8
36. 36.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 48 (c) 49 (d) 47
37. 37.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 169 (c) 97 (d) 25
38. 38.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
39. 39.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)
40. 40.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)

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