Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 336 · 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 24²? (a) 256 (b) 586 (c) 416 (d) 576
2. 2.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 185 (c) 273 (d) 361
3. 3.If a + b = 6 and ab = 5, what is (a − b)²? (a) 16 (b) 4 (c) 26 (d) 36
4. 4.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)
5. 5.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 11 (c) 13 (d) 12
6. 6.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 9216 (c) 10826 (d) 10016
7. 7.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 2 and 9 (c) 3 and 6 (d) 1 and 8
8. 8.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1849 (c) 1369 (d) 1609
9. 9.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x − 3)(x − 7) (c) (x + 1)(x + 9) (d) (x + 3)(x + 7)
10. 10.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 266 (c) 384 (d) 256
11. 11.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)
12. 12.If a + b = 17 and ab = 66, what is (a − b)²? (a) 25 (b) 5 (c) 157 (d) 289
13. 13.If a + b = 8 and ab = 15, what is (a − b)²? (a) 4 (b) 64 (c) 34 (d) 2
14. 14.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 56 (d) 55
15. 15.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
16. 16.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 17 (c) 10 (d) 16
17. 17.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
18. 18.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x − 4)(x − 5)
19. 19.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
20. 20.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 69 (c) 29 (d) 39
21. 21.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
22. 22.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 46 (c) 44 (d) 45
23. 23.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 64 (c) 160 (d) 8
24. 24.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 10 (c) 11 (d) 24
25. 25.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
26. 26.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 2025 (c) 3025 (d) 2525
27. 27.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 909 (c) 1099 (d) 1089
28. 28.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 4 and 8 (c) 1 and 32 (d) 1 and 11
29. 29.If a + b = 5 and ab = 6, what is (a − b)²? (a) 1 (b) 13 (c) 25 (d) 19
30. 30.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 8649 (c) 8109 (d) 7569
31. 31.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 433 (c) 145 (d) 289
32. 32.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 1 (c) 225 (d) 169
33. 33.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1596 (c) 1454 (d) 1764
34. 34.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 15 (d) 63
35. 35.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 271 (d) 541
36. 36.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 63 (c) 117 (d) 81
37. 37.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x + 1)(x + 27)
38. 38.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4634 (c) 4896 (d) 4624
39. 39.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 494 (d) 484
40. 40.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 10 (c) 8 (d) 9

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