Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 952 · 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 21²? (a) 361 (b) 401 (c) 451 (d) 441
2. 2.If a + b = 5 and ab = 6, what is (a − b)²? (a) 1 (b) 25 (c) 13 (d) 19
3. 3.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 3025 (c) 2035 (d) 2025
4. 4.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
5. 5.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)
6. 6.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 11 (c) 12 (d) 13
7. 7.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4771 (c) 4899 (d) 5041
8. 8.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 127 (c) 85 (d) 169
9. 9.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 1 and 4 (c) 2 and 4 (d) 2 and 3
10. 10.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 117 (c) 144 (d) 90
11. 11.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
12. 12.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 100 (c) 79 (d) 58
13. 13.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
14. 14.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 9)
15. 15.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 25 (c) 33 (d) 17
16. 16.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 9991 (c) 9419 (d) 10609
17. 17.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x − 3)(x − 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
18. 18.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 3 (c) 149 (d) 289
19. 19.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x − 2)(x − 4) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
20. 20.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 5 (c) 9 (d) 7
21. 21.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 1024 (c) 784 (d) 904
22. 22.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
23. 23.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 1 and 32 (c) 4 and 8 (d) 2 and 16
24. 24.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 4761 (c) 5051 (d) 4901
25. 25.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 2 and 8 (c) 3 and 7 (d) 1 and 21
26. 26.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 8836 (c) 7406 (d) 7396
27. 27.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 49 (c) 73 (d) 37
28. 28.If a + b = 8 and ab = 12, what is a² + b²? (a) 40 (b) 52 (c) 64 (d) 88
29. 29.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 11 (c) 21 (d) 9
30. 30.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 1225 (c) 2035 (d) 2025
31. 31.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8291 (c) 8281 (d) 7921
32. 32.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
33. 33.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
34. 34.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 529 (c) 193 (d) 277
35. 35.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9614 (c) 9604 (d) 10404
36. 36.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 193 (c) 5 (d) 361
37. 37.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 85 (c) 127 (d) 1
38. 38.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
39. 39.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 13 (c) 14 (d) 12
40. 40.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 3 (c) 185 (d) 361

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